Red Wine Exploration by Krushab Gandhi


In this project, we will explore a data set on wine quality. The dataset was obatined from the UCI Machine Learning Repository. The objective is to explore which chemical properties influence the quality of red wines.

We will start by exploring the data using the statistical program, R. As interesting relationships in the data are discovered, we will produce and refine plots to illustrate them.


Firstly, we will load the Wine dataset and analyze its structure

Wine <- read.csv('~/Desktop/Resume/Code-and-Dataset/wineQualityReds.csv')
str(Wine)
## 'data.frame':    1599 obs. of  13 variables:
##  $ X                   : int  1 2 3 4 5 6 7 8 9 10 ...
##  $ fixed.acidity       : num  7.4 7.8 7.8 11.2 7.4 7.4 7.9 7.3 7.8 7.5 ...
##  $ volatile.acidity    : num  0.7 0.88 0.76 0.28 0.7 0.66 0.6 0.65 0.58 0.5 ...
##  $ citric.acid         : num  0 0 0.04 0.56 0 0 0.06 0 0.02 0.36 ...
##  $ residual.sugar      : num  1.9 2.6 2.3 1.9 1.9 1.8 1.6 1.2 2 6.1 ...
##  $ chlorides           : num  0.076 0.098 0.092 0.075 0.076 0.075 0.069 0.065 0.073 0.071 ...
##  $ free.sulfur.dioxide : num  11 25 15 17 11 13 15 15 9 17 ...
##  $ total.sulfur.dioxide: num  34 67 54 60 34 40 59 21 18 102 ...
##  $ density             : num  0.998 0.997 0.997 0.998 0.998 ...
##  $ pH                  : num  3.51 3.2 3.26 3.16 3.51 3.51 3.3 3.39 3.36 3.35 ...
##  $ sulphates           : num  0.56 0.68 0.65 0.58 0.56 0.56 0.46 0.47 0.57 0.8 ...
##  $ alcohol             : num  9.4 9.8 9.8 9.8 9.4 9.4 9.4 10 9.5 10.5 ...
##  $ quality             : int  5 5 5 6 5 5 5 7 7 5 ...

Let’s see the summary of the Wine dataset

summary(Wine)
##        X          fixed.acidity   volatile.acidity  citric.acid   
##  Min.   :   1.0   Min.   : 4.60   Min.   :0.1200   Min.   :0.000  
##  1st Qu.: 400.5   1st Qu.: 7.10   1st Qu.:0.3900   1st Qu.:0.090  
##  Median : 800.0   Median : 7.90   Median :0.5200   Median :0.260  
##  Mean   : 800.0   Mean   : 8.32   Mean   :0.5278   Mean   :0.271  
##  3rd Qu.:1199.5   3rd Qu.: 9.20   3rd Qu.:0.6400   3rd Qu.:0.420  
##  Max.   :1599.0   Max.   :15.90   Max.   :1.5800   Max.   :1.000  
##  residual.sugar     chlorides       free.sulfur.dioxide
##  Min.   : 0.900   Min.   :0.01200   Min.   : 1.00      
##  1st Qu.: 1.900   1st Qu.:0.07000   1st Qu.: 7.00      
##  Median : 2.200   Median :0.07900   Median :14.00      
##  Mean   : 2.539   Mean   :0.08747   Mean   :15.87      
##  3rd Qu.: 2.600   3rd Qu.:0.09000   3rd Qu.:21.00      
##  Max.   :15.500   Max.   :0.61100   Max.   :72.00      
##  total.sulfur.dioxide    density             pH          sulphates     
##  Min.   :  6.00       Min.   :0.9901   Min.   :2.740   Min.   :0.3300  
##  1st Qu.: 22.00       1st Qu.:0.9956   1st Qu.:3.210   1st Qu.:0.5500  
##  Median : 38.00       Median :0.9968   Median :3.310   Median :0.6200  
##  Mean   : 46.47       Mean   :0.9967   Mean   :3.311   Mean   :0.6581  
##  3rd Qu.: 62.00       3rd Qu.:0.9978   3rd Qu.:3.400   3rd Qu.:0.7300  
##  Max.   :289.00       Max.   :1.0037   Max.   :4.010   Max.   :2.0000  
##     alcohol         quality     
##  Min.   : 8.40   Min.   :3.000  
##  1st Qu.: 9.50   1st Qu.:5.000  
##  Median :10.20   Median :6.000  
##  Mean   :10.42   Mean   :5.636  
##  3rd Qu.:11.10   3rd Qu.:6.000  
##  Max.   :14.90   Max.   :8.000


Converting the X identifier into factors and quality attribute into numeric

Wine$quality <- as.numeric(Wine$quality)
Wine$X <- as.factor(Wine$X)

We use the table() function to calculate how many wines we have for each quality

table(Wine$quality)
## 
##   3   4   5   6   7   8 
##  10  53 681 638 199  18

UNIVARIATE PLOTS SECTION
Analyzing each and every attribute in the Wine dataset.

library(ggplot2)
library(gridExtra)
grid.arrange(qplot(Wine$fixed.acidity, geom = "histogram",  xlab = "Fixed Acidity", fill=I("blue"),
                   col=I("red")),
             qplot(Wine$volatile.acidity, geom = "histogram", xlab = "Volatile Acidity", fill=I("blue"),
                   col=I("red")),
             qplot(Wine$citric.acid, geom = "histogram", xlab = "Citric Acid", fill=I("blue"),
                   col=I("red")),
             qplot(Wine$residual.sugar, geom = "histogram", xlab = "Residual Sugar", fill=I("blue"),
                   col=I("red")),
             qplot(Wine$chlorides, geom = "histogram", binwidth = 0.03, xlab = "Chlorides", fill=I("blue"), 
                   col=I("red")),
             qplot(Wine$free.sulfur.dioxide, geom = "histogram", xlab = "Free Sulfur Dioxide", 
                   fill=I("blue"), col=I("red")),
             qplot(Wine$total.sulfur.dioxide, geom = "histogram", xlab = "Total Sulfur Dioxide",
                   fill=I("blue"), col=I("red")),
             qplot(Wine$density, geom = "histogram", xlab = "Density", fill=I("blue"),
                   col=I("red") ), 
             qplot(Wine$pH, geom = "histogram", xlab = "pH", fill=I("blue"),
                   col=I("red")),
             qplot(Wine$sulphates, geom = "histogram", xlab = "Sulphates", fill=I("blue"),
                   col=I("red")),
             qplot(Wine$alcohol, geom = "histogram", xlab = "Alcohol", fill=I("blue"), 
                   col=I("red")),
             qplot(Wine$quality, geom = "histogram", xlab = "Quality", fill=I("blue"), 
                   col=I("red")),
             ncol = 4, bottom = "Univariate Analysis - Histograms")


UNIVARIATE ANALYSIS
From the plots, we come to know the following:


Wine$rating <- ifelse(Wine$quality < 5, 'bad', ifelse(Wine$quality < 7, 'average', 'good'))
Wine$rating <- ordered(Wine$rating, levels = c('bad', 'average', 'good'))
summary(Wine$rating)
##     bad average    good 
##      63    1319     217
table(Wine$rating)
## 
##     bad average    good 
##      63    1319     217

To explore better, we created a new variable Rating, which classifies Wines into Good, Bad, and Average depending on their quality scores.
- BAD is Quality less than 5.
- AVERAGE is Quality less than 7 but greater than or equal to 5.
- GOOD is any number above and including 7.


ggplot(Wine,aes(x=fixed.acidity)) + geom_histogram(fill='red') + scale_x_log10(breaks=1:15) + 
       xlab('Fixed Acidity') + ylab('Count') + ggtitle('Histogram of Fixed Acidity Values') + 
       theme(plot.title = element_text(hjust=0.5))

ggplot(Wine) + geom_histogram(aes(x=volatile.acidity),fill='blue') + scale_x_log10(breaks=seq(0.1,1,0.1)) + 
  ggtitle("Histogram for volatile acidity") + theme(plot.title = element_text(hjust=0.5))

ggplot(Wine) + geom_histogram(aes(x=citric.acid), fill='green') + scale_x_log10() + 
  ggtitle("Histogram for citric acid") + theme(plot.title = element_text(hjust=0.5))



library(plotly)
Wine$TAC.acidity <- Wine$fixed.acidity + Wine$volatile.acidity + Wine$citric.acid
qplot(Wine$TAC.acidity,main = 'Histogram of TAC Acidity (Tartaric+Acetic+Citric)')

p <- plot_ly(Wine,x=~TAC.acidity) %>% layout(xaxis = list(title = "Histogram of TAC Acidity using plolty"))
p


get_simple_boxplot <- function(column, ylab) {
return(qplot(data = Wine, x = '',
y = column, geom = 'boxplot',
xlab = '',
ylab = ylab))
}

grid.arrange(get_simple_boxplot(Wine$fixed.acidity, 'fixed acidity'),
get_simple_boxplot(Wine$volatile.acidity, 'volatile acidity'),
get_simple_boxplot(Wine$citric.acid, 'citric acid'),
get_simple_boxplot(Wine$TAC.acidity, 'TAC acidity'),
get_simple_boxplot(Wine$residual.sugar, 'residual sugar'),
get_simple_boxplot(Wine$chlorides, 'chlorides'),
get_simple_boxplot(Wine$free.sulfur.dioxide, 'free sulf. dioxide'),
get_simple_boxplot(Wine$total.sulfur.dioxide, 'total sulf. dioxide'),
get_simple_boxplot(Wine$density, 'density'),
get_simple_boxplot(Wine$pH, 'pH'),
get_simple_boxplot(Wine$sulphates, 'sulphates'),
get_simple_boxplot(Wine$alcohol, 'alcohol'),
ncol = 4)



plot_ly(Wine,y=~alcohol,type='box')

BIVARIATE PLOTS SECTION


set.seed(1)
Wine_sample <- Wine[,-which(names(Wine) %in% c('X', 'rating'))][sample(1:length(Wine$quality), 40), ]

get_bivariate_boxplot <- function(x, y, ylab) {
return(qplot(data = Wine, x = x, y = y, geom = 'boxplot', ylab = ylab))
}

grid.arrange(get_bivariate_boxplot(Wine$quality, Wine$fixed.acidity,
'fixed acidity'),
get_bivariate_boxplot(Wine$quality, Wine$volatile.acidity,
'volatile acidity'),
get_bivariate_boxplot(Wine$quality, Wine$citric.acid,
'citric acid'),
get_bivariate_boxplot(Wine$quality, Wine$TAC.acidity,
'TAC acidity'),
get_bivariate_boxplot(Wine$quality, log10(Wine$residual.sugar),
'residual sugar'),
get_bivariate_boxplot(Wine$quality, log10(Wine$chlorides),
'chlorides'),
get_bivariate_boxplot(Wine$quality, Wine$free.sulfur.dioxide,
'free sulf. dioxide'),
get_bivariate_boxplot(Wine$quality, Wine$total.sulfur.dioxide,
'total sulf. dioxide'),
get_bivariate_boxplot(Wine$quality, Wine$density,
'density'),
get_bivariate_boxplot(Wine$quality, Wine$pH,
'pH'),
get_bivariate_boxplot(Wine$quality, log10(Wine$sulphates),
'sulphates'),
get_bivariate_boxplot(Wine$quality, Wine$alcohol,
'alcohol'),
ncol = 4, bottom = "Here 'x' is quality")

grid.arrange(get_bivariate_boxplot(Wine$rating, Wine$fixed.acidity,
'fixed acidity'),
get_bivariate_boxplot(Wine$rating, Wine$volatile.acidity,
'volatile acidity'),
get_bivariate_boxplot(Wine$rating, Wine$citric.acid,
'citric acid'),
get_bivariate_boxplot(Wine$rating, Wine$TAC.acidity,
'TAC acidity'),
get_bivariate_boxplot(Wine$rating, log10(Wine$residual.sugar),
'residual sugar'),
get_bivariate_boxplot(Wine$rating, log10(Wine$chlorides),
'chlorides'),
get_bivariate_boxplot(Wine$rating, Wine$free.sulfur.dioxide,
'free sulf. dioxide'),
get_bivariate_boxplot(Wine$rating, Wine$total.sulfur.dioxide,
'total sulf. dioxide'),
get_bivariate_boxplot(Wine$rating, Wine$density,
'density'),
get_bivariate_boxplot(Wine$rating, Wine$pH,
'pH'),
get_bivariate_boxplot(Wine$rating, log10(Wine$sulphates),
'sulphates'),
get_bivariate_boxplot(Wine$rating, Wine$alcohol,
'alcohol'),
ncol = 4, bottom = "Here 'x' is rating")


Example using Plotly


plot_ly(Wine,x=~quality,y=~alcohol)

Results from Bivariate Analysis are as follows:

From exploring these plots, it seems that a ‘good’ wine generally has these trends: - Higher fixed acidity (tartaric acid) and citric acid
- Lower volatile acidity (acetic acid)
- Lower pH (i.e. more acidic)
- Higher sulphates
- Higher alcohol
- To a lesser extent, lower chlorides and lower density
- Residual sugar and sulfur dioxides did not seem to have a dramatic impact on the quality or rating of the wines.
- Interestingly, it appears that different types of acid affect wine quality different; as such, TAC.acidity saw an attenuated trend, as the presence of volatile (acetic) acid accompanied decreased quality.


simple_cor_test <- function(x, y) {
return(cor.test(x, as.numeric(y))$estimate)
}

correlations <- c(
simple_cor_test(Wine$fixed.acidity, Wine$quality),
simple_cor_test(Wine$volatile.acidity, Wine$quality),
simple_cor_test(Wine$citric.acid, Wine$quality),
simple_cor_test(Wine$TAC.acidity, Wine$quality),
simple_cor_test(log10(Wine$residual.sugar), Wine$quality),
simple_cor_test(log10(Wine$chlorides), Wine$quality),
simple_cor_test(Wine$free.sulfur.dioxide, Wine$quality),
simple_cor_test(Wine$total.sulfur.dioxide, Wine$quality),
simple_cor_test(Wine$density, Wine$quality),
simple_cor_test(Wine$pH, Wine$quality),
simple_cor_test(log10(Wine$sulphates), Wine$quality),
simple_cor_test(Wine$alcohol, Wine$quality))

correlations
##         cor         cor         cor         cor         cor         cor 
##  0.12405165 -0.39055778  0.22637251  0.10375373  0.02353331 -0.17613996 
##         cor         cor         cor         cor         cor         cor 
## -0.05065606 -0.18510029 -0.17491923 -0.05773139  0.30864193  0.47616632
names(correlations) <- c('fixed.acidity', 'volatile.acidity', 'citric.acid',
'TAC.acidity', 'log10.residual.sugar',
'log10.chlordies', 'free.sulfur.dioxide',
'total.sulfur.dioxide', 'density', 'pH',
'log10.sulphates', 'alcohol')
correlations
##        fixed.acidity     volatile.acidity          citric.acid 
##           0.12405165          -0.39055778           0.22637251 
##          TAC.acidity log10.residual.sugar      log10.chlordies 
##           0.10375373           0.02353331          -0.17613996 
##  free.sulfur.dioxide total.sulfur.dioxide              density 
##          -0.05065606          -0.18510029          -0.17491923 
##                   pH      log10.sulphates              alcohol 
##          -0.05773139           0.30864193           0.47616632

By utilizing cor.test, I calculated the correlation for each of these variables against quality.

As we see, the top 4 come to be: - Alcohol
- Sulphates (log10)
- Volatile acidity
- Citric acid

Let’s plot them against each other using RATING as a facet.


ggplot(data=Wine,aes(x=citric.acid,y=alcohol))+facet_wrap(~rating)+geom_jitter(alpha=0.2)

ggplot(data = Wine, aes(x = log10(sulphates), y = alcohol)) +
  facet_wrap(~rating) +
  geom_jitter(alpha=0.2) 

ggplot(data = Wine, aes(x = volatile.acidity, y = alcohol)) +
  facet_wrap(~rating) +
  geom_point(alpha=0.2)

ggplot(data = Wine, aes(x = citric.acid, y = alcohol)) +
  facet_wrap(~rating) +
  geom_point(alpha=0.2)

ggplot(data = Wine, aes(x = volatile.acidity, y = log10(sulphates))) +
  facet_wrap(~rating) +
  geom_jitter(alpha=0.2)

ggplot(data = Wine, aes(x = citric.acid, y = log10(sulphates))) +
  facet_wrap(~rating) +
  geom_point(alpha=0.2)

ggplot(data = Wine, aes(x = citric.acid, y = volatile.acidity)) +
  facet_wrap(~rating) +
  geom_point(alpha=0.2)


Lets examining the acidity variables:


ggplot(data = Wine, aes(x = fixed.acidity, y = citric.acid)) +
  geom_point(alpha=0.3)

  cor.test(Wine$fixed.acidity, Wine$citric.acid)
## 
##  Pearson's product-moment correlation
## 
## data:  Wine$fixed.acidity and Wine$citric.acid
## t = 36.234, df = 1597, p-value < 2.2e-16
## alternative hypothesis: true correlation is not equal to 0
## 95 percent confidence interval:
##  0.6438839 0.6977493
## sample estimates:
##       cor 
## 0.6717034
plot_ly(Wine,x=~fixed.acidity,y=~citric.acid)
ggplot(data = Wine, aes(x = volatile.acidity, y = citric.acid)) +
  geom_point(alpha=0.3)

cor.test(Wine$volatile.acidity, Wine$citric.acid)
## 
##  Pearson's product-moment correlation
## 
## data:  Wine$volatile.acidity and Wine$citric.acid
## t = -26.489, df = 1597, p-value < 2.2e-16
## alternative hypothesis: true correlation is not equal to 0
## 95 percent confidence interval:
##  -0.5856550 -0.5174902
## sample estimates:
##        cor 
## -0.5524957
ggplot(data = Wine, aes(x = log10(TAC.acidity), y = pH)) +
  geom_point(alpha=0.3)

cor.test(log10(Wine$TAC.acidity), Wine$pH)
## 
##  Pearson's product-moment correlation
## 
## data:  log10(Wine$TAC.acidity) and Wine$pH
## t = -39.663, df = 1597, p-value < 2.2e-16
## alternative hypothesis: true correlation is not equal to 0
## 95 percent confidence interval:
##  -0.7283140 -0.6788653
## sample estimates:
##        cor 
## -0.7044435

Upon examining, I observed strong correlations between the acidity variables.

An interesting question to pose, using basic chemistry knowledge, is to ask what other components other than the measured acids are affecting pH. We can quantify this difference by building a predictive linear model, to predict pH based on TAC.acidity and capture the % difference as a new variable.


# Linear model
m <- lm(I(pH) ~ I(log10(TAC.acidity)), data = Wine)
Wine$pH.predictions <- predict(m, Wine)
predict(m,Wine)
##        1        2        3        4        5        6        7        8 
## 3.367560 3.328355 3.333604 3.142857 3.367560 3.370367 3.336247 3.378157 
##        9       10       11       12       13       14       15       16 
## 3.346943 3.349649 3.421872 3.349649 3.517732 3.327050 3.265369 3.264785 
##       17       18       19       20       21       22       23       24 
## 3.286809 3.311623 3.369664 3.325098 3.271244 3.353733 3.337573 3.301567 
##       25       26       27       28       29       30       31       32 
## 3.415743 3.462582 3.357158 3.337573 3.388229 3.343914 3.415362 3.404801 
##       33       34       35       36       37       38       39       40 
## 3.303126 3.401819 3.559849 3.343914 3.337573 3.323154 3.456818 3.366861 
##       41       42       43       44       45       46       47       48 
## 3.366861 3.264785 3.361296 3.309092 3.411946 3.611234 3.303752 3.276584 
##       49       50       51       52       53       54       55       56 
## 3.447877 3.511834 3.264202 3.437490 3.439076 3.286809 3.356471 3.349649 
##       57       58       59       60       61       62       63       64 
## 3.184362 3.357158 3.335585 3.374603 3.271244 3.315441 3.361989 3.390047 
##       65       66       67       68       69       70       71       72 
## 3.376377 3.376377 3.365464 3.420718 3.237432 3.323477 3.346269 3.333604 
##       73       74       75       76       77       78       79       80 
## 3.332945 3.293219 3.217212 3.256670 3.256670 3.404801 3.405923 3.300012 
##       81       82       83       84       85       86       87       88 
## 3.462582 3.312258 3.348972 3.358534 3.443860 3.403681 3.284991 3.343579 
##       89       90       91       92       93       94       95       96 
## 3.240780 3.396270 3.328355 3.284991 3.284387 3.343579 3.532049 3.590118 
##       97       98       99      100      101      102      103      104 
## 3.405549 3.392601 3.336247 3.318963 3.294755 3.333604 3.318963 3.314484 
##      105      106      107      108      109      110      111      112 
## 3.379585 3.314484 3.314803 3.439076 3.316719 3.282878 3.336909 3.300944 
##      113      114      115      116      117      118      119      120 
## 3.301567 3.201854 3.336909 3.241900 3.300322 3.341570 3.283783 3.391140 
##      121      122      123      124      125      126      127      128 
## 3.342908 3.283783 3.374957 3.326399 3.342239 3.267712 3.275393 3.281373 
##      129      130      131      132      133      134      135      136 
## 3.323801 3.481088 3.288938 3.520017 3.520017 3.441463 3.308461 3.291992 
##      137      138      139      140      141      142      143      144 
## 3.297532 3.376377 3.336909 3.336909 3.291992 3.297532 3.582909 3.469242 
##      145      146      147      148      149      150      151      152 
## 3.582909 3.288025 3.492314 3.350328 3.411946 3.305317 3.367560 3.208687 
##      153      154      155      156      157      158      159      160 
## 3.365464 3.365464 3.378157 3.378157 3.378157 3.378157 3.390411 3.405175 
##      161      162      163      164      165      166      167      168 
## 3.334924 3.353733 3.348972 3.356471 3.364072 3.313529 3.408174 3.383171 
##      169      170      171      172      173      174      175      176 
## 3.411190 3.343914 3.319606 3.334263 3.334263 3.369664 3.382452 3.415743 
##      177      178      179      180      181      182      183      184 
## 3.382452 3.344250 3.388592 3.274204 3.274204 3.248102 3.378157 3.402935 
##      185      186      187      188      189      190      191      192 
## 3.408927 3.260713 3.355785 3.339901 3.325098 3.326399 3.304690 3.448684 
##      193      194      195      196      197      198      199      200 
## 3.407423 3.349649 3.349649 3.325748 3.361989 3.126155 3.508683 3.371070 
##      201      202      203      204      205      206      207      208 
## 3.226413 3.268299 3.418034 3.391140 3.389683 3.063871 3.063871 3.328355 
##      209      210      211      212      213      214      215      216 
## 3.338902 3.150441 3.202900 3.310040 3.113496 3.305945 3.332616 3.376022 
##      217      218      219      220      221      222      223      224 
## 3.278076 3.305004 3.361989 3.329662 3.339568 3.361296 3.414982 3.277479 
##      225      226      227      228      229      230      231      232 
## 3.283481 3.348295 3.248669 3.250374 3.348295 3.398483 3.564783 3.344250 
##      233      234      235      236      237      238      239      240 
## 3.303439 3.398483 3.289852 3.386779 3.386779 3.385694 3.386779 3.289852 
##      241      242      243      244      245      246      247      248 
## 3.253513 3.101990 3.348295 2.994194 2.994194 3.377444 3.385333 3.309724 
##      249      250      251      252      253      254      255      256 
## 3.354416 3.377444 3.165920 3.396270 3.148060 3.314484 3.396270 3.320571 
##      257      258      259      260      261      262      263      264 
## 3.241900 3.426901 3.316079 3.205523 3.342908 3.373541 3.336909 3.340234 
##      265      266      267      268      269      270      271      272 
## 3.075876 3.117985 3.309724 3.326399 3.412703 3.135838 3.339901 3.135838 
##      273      274      275      276      277      278      279      280 
## 3.151874 3.288329 3.351687 3.339901 3.412703 3.135838 3.190474 3.270064 
##      281      282      283      284      285      286      287      288 
## 3.140039 3.330317 3.357846 3.270064 3.217212 3.217212 3.099367 3.373187 
##      289      290      291      292      293      294      295      296 
## 3.288633 3.119338 3.288633 3.160066 3.184869 3.410434 3.050913 3.156196 
##      297      298      299      300      301      302      303      304 
## 3.154271 3.389683 3.383890 3.401448 3.368261 3.156196 3.300012 3.361296 
##      305      306      307      308      309      310      311      312 
## 3.268299 3.178315 3.337573 3.187410 3.185376 3.369664 3.178315 3.329008 
##      313      314      315      316      317      318      319      320 
## 3.261293 3.284991 3.371070 3.393333 3.226413 3.220983 3.201854 3.220983 
##      321      322      323      324      325      326      327      328 
## 3.201854 3.238546 3.342239 3.178315 3.210276 3.210276 3.108599 3.181835 
##      329      330      331      332      333      334      335      336 
## 3.045732 3.166411 3.183856 3.183856 3.316719 3.305317 3.336247 3.093942 
##      337      338      339      340      341      342      343      344 
## 3.260713 3.336909 3.079233 3.085581 3.097625 3.168871 3.156196 3.156196 
##      345      346      347      348      349      350      351      352 
## 3.100677 3.397376 3.416124 3.019757 3.222065 3.254659 3.164451 3.254086 
##      353      354      355      356      357      358      359      360 
## 3.349310 3.025087 3.474288 3.414221 3.124785 3.164941 3.099803 3.072123 
##      361      362      363      364      365      366      367      368 
## 3.299701 3.285597 3.093295 3.074205 3.054326 3.198208 3.054326 3.164696 
##      369      370      371      372      373      374      375      376 
## 3.185884 3.236876 3.397376 3.337573 3.256095 3.362682 3.016733 3.110375 
##      377      378      379      380      381      382      383      384 
## 3.123873 3.236876 3.113273 3.301567 3.309092 3.026044 3.309092 3.309092 
##      385      386      387      388      389      390      391      392 
## 3.340902 3.367560 3.333604 3.300944 3.338902 3.231896 3.492314 3.026044 
##      393      394      395      396      397      398      399      400 
## 3.226413 3.266832 3.059383 3.093726 3.422257 3.113496 3.113496 3.266247 
##      401      402      403      404      405      406      407      408 
## 3.422257 3.356471 3.089854 3.122055 3.344250 3.278973 3.209217 3.097191 
##      409      410      411      412      413      414      415      416 
## 3.177814 3.080075 3.261293 3.253800 3.374957 3.202900 3.267712 3.273315 
##      417      418      419      420      421      422      423      424 
## 3.158128 3.396270 3.108599 3.405923 3.226413 3.413462 3.316719 3.183350 
##      425      426      427      428      429      430      431      432 
## 3.316719 3.413462 3.438282 3.220443 3.250944 3.046526 3.183350 3.327050 
##      433      434      435      436      437      438      439      440 
## 3.101114 3.085581 3.175814 3.085581 3.309724 3.131668 3.175814 3.388955 
##      441      442      443      444      445      446      447      448 
## 3.072539 3.101552 2.945789 3.197689 3.549143 3.229422 3.078812 3.244147 
##      449      450      451      452      453      454      455      456 
## 3.286203 3.100240 3.100240 3.289242 3.419566 3.175814 3.401448 3.117534 
##      457      458      459      460      461      462      463      464 
## 3.254946 3.245839 3.175814 3.106388 3.239662 3.299391 3.147585 3.279572 
##      465      466      467      468      469      470      471      472 
## 3.128216 3.210276 3.182844 3.264785 3.123873 3.274798 3.058571 3.217212 
##      473      474      475      476      477      478      479      480 
## 3.081341 3.204473 3.179318 3.219364 3.249237 3.187410 3.219364 3.237154 
##      481      482      483      484      485      486      487      488 
## 3.180324 3.233551 3.166411 3.165920 3.158128 3.180827 3.180827 3.183603 
##      489      490      491      492      493      494      495      496 
## 3.122964 3.243023 3.228874 3.241900 3.258977 3.265369 3.440666 3.164941 
##      497      498      499      500      501      502      503      504 
## 3.335585 3.384611 3.164941 3.265369 3.335585 3.165430 3.165430 3.182339 
##      505      506      507      508      509      510      511      512 
## 3.185884 3.177814 3.188940 3.125241 3.199248 3.043752 3.086007 3.199248 
##      513      514      515      516      517      518      519      520 
## 3.164941 3.161524 3.161524 3.268593 3.074205 3.168871 3.163962 3.363724 
##      521      522      523      524      525      526      527      528 
## 3.218287 3.340234 3.302814 3.237432 3.241340 3.179821 3.363724 3.383890 
##      529      530      531      532      533      534      535      536 
## 3.300944 3.206049 3.273611 3.109487 3.109487 3.203948 3.208687 3.273611 
##      537      538      539      540      541      542      543      544 
## 3.206049 3.297532 3.067982 3.124329 3.246404 3.223149 3.233827 3.140039 
##      545      546      547      548      549      550      551      552 
## 3.005167 3.244711 3.354416 3.173822 3.088997 3.246969 3.424961 3.244147 
##      553      554      555      556      557      558      559      560 
## 3.236876 3.511834 2.959592 2.959592 3.148535 2.956194 3.148535 3.058977 
##      561      562      563      564      565      566      567      568 
## 3.065923 3.252084 3.246404 3.348295 3.058977 3.065923 3.268887 3.268887 
##      569      570      571      572      573      574      575      576 
## 3.204998 3.466735 3.128905 3.466735 3.197689 3.164941 3.172332 3.096757 
##      577      578      579      580      581      582      583      584 
## 3.212934 3.263619 3.261874 3.173822 3.086860 3.086860 3.113496 3.109487 
##      585      586      587      588      589      590      591      592 
## 3.116185 3.350328 3.149487 3.355785 3.570761 3.195102 3.251514 3.412703 
##      593      594      595      596      597      598      599      600 
## 3.251514 3.195619 3.321216 3.253513 3.086007 3.096757 3.291380 3.067982 
##      601      602      603      604      605      606      607      608 
## 3.284085 3.050113 3.337905 3.050113 3.307202 3.305945 3.231896 3.265954 
##      609      610      611      612      613      614      615      616 
## 3.187919 3.462582 3.272426 3.052115 3.364768 3.309724 3.240500 3.209746 
##      617      618      619      620      621      622      623      624 
## 3.209746 3.129824 3.127986 3.140977 3.302814 3.308461 3.204473 3.329662 
##      625      626      627      628      629      630      631      632 
## 3.411946 3.411946 3.265954 3.265954 3.277180 3.339235 3.277180 3.182844 
##      633      634      635      636      637      638      639      640 
## 3.364072 3.181079 3.342908 3.274798 3.206576 3.212135 3.324774 3.274798 
##      641      642      643      644      645      646      647      648 
## 3.199768 3.218825 3.199768 3.218825 3.199768 3.337573 3.373187 3.298151 
##      649      650      651      652      653      654      655      656 
## 3.278375 3.419566 3.167885 3.197689 2.950297 3.230245 3.274798 3.224778 
##      657      658      659      660      661      662      663      664 
## 3.167885 3.095456 3.389683 3.377444 3.389683 3.358534 3.386779 3.202377 
##      665      666      667      668      669      670      671      672 
## 3.098496 3.240780 3.298461 3.140507 3.185884 3.140507 3.408174 3.299080 
##      673      674      675      676      677      678      679      680 
## 3.174817 3.299080 3.163474 3.242461 3.163474 3.276584 3.296605 3.168378 
##      681      682      683      684      685      686      687      688 
## 3.044939 3.328355 3.291074 3.300944 3.188940 3.300944 3.379585 3.249805 
##      689      690      691      692      693      694      695      696 
## 3.346943 3.310356 3.334593 3.228052 3.271244 3.259555 3.260713 3.577816 
##      697      698      699      700      701      702      703      704 
## 3.398483 3.398483 3.231620 3.111265 3.142386 3.398483 3.399223 3.349649 
##      705      706      707      708      709      710      711      712 
## 3.253513 3.272131 3.384611 3.368962 3.342573 3.221524 3.142151 3.259266 
##      713      714      715      716      717      718      719      720 
## 3.283783 3.321216 3.198208 3.383890 3.321216 3.362682 3.307831 3.391870 
##      721      722      723      724      725      726      727      728 
## 3.307831 3.270064 3.367560 3.395535 3.326073 3.257822 3.313529 3.451112 
##      729      730      731      732      733      734      735      736 
## 3.451112 3.426901 3.209217 3.253513 3.363029 3.389683 3.342908 3.322830 
##      737      738      739      740      741      742      743      744 
## 3.322830 3.321216 3.265954 3.265954 3.286203 3.249805 3.441064 3.117534 
##      745      746      747      748      749      750      751      752 
## 3.143328 3.375312 3.312893 3.287417 3.383171 3.375312 3.304690 3.304690 
##      753      754      755      756      757      758      759      760 
## 3.355785 3.304690 3.310356 3.321861 3.427289 3.309724 3.309724 3.281373 
##      761      762      763      764      765      766      767      768 
## 3.257822 3.236043 3.277180 3.236043 3.254372 3.249805 3.273018 3.345595 
##      769      770      771      772      773      774      775      776 
## 3.395535 3.331630 3.395535 3.228874 3.229148 3.334263 3.333604 3.360604 
##      777      778      779      780      781      782      783      784 
## 3.385694 3.395902 3.305945 3.399963 3.405923 3.442261 3.255520 3.442261 
##      785      786      787      788      789      790      791      792 
## 3.396270 3.211869 3.211869 3.204473 3.204473 3.283179 3.360604 3.270654 
##      793      794      795      796      797      798      799      800 
## 3.394066 3.353733 3.198728 3.145216 3.278973 3.241900 3.234657 3.234657 
##      801      802      803      804      805      806      807      808 
## 3.382452 3.292912 3.568262 3.351007 3.299080 3.315441 3.305317 3.315441 
##      809      810      811      812      813      814      815      816 
## 3.371070 3.347619 3.382452 3.059383 3.164941 3.408927 3.075876 3.164941 
##      817      818      819      820      821      822      823      824 
## 3.220443 3.168378 3.387866 3.253227 3.397376 3.605881 3.421102 3.421102 
##      825      826      827      828      829      830      831      832 
## 3.384611 3.396270 3.366861 3.396270 3.342908 3.484518 3.356815 3.484518 
##      833      834      835      836      837      838      839      840 
## 3.180827 3.121148 3.262745 3.349310 3.429627 3.429627 3.206576 3.488836 
##      841      842      843      844      845      846      847      848 
## 3.145216 3.429627 3.166411 3.365116 3.214535 3.430409 3.430409 3.357846 
##      849      850      851      852      853      854      855      856 
## 3.430409 3.431191 3.238546 3.238546 3.324449 3.245274 3.245274 3.349988 
##      857      858      859      860      861      862      863      864 
## 3.245274 3.320571 3.123873 3.405175 3.383171 3.413462 3.357846 3.383171 
##      865      866      867      868      869      870      871      872 
## 3.383171 3.381375 3.410434 3.400705 3.405175 3.356471 3.345259 3.411946 
##      873      874      875      876      877      878      879      880 
## 3.382452 3.266539 3.181835 3.276584 3.405923 3.345259 3.271244 3.386056 
##      881      882      883      884      885      886      887      888 
## 3.251514 3.352709 3.307831 3.386056 3.271244 3.260134 3.252655 3.163962 
##      889      890      891      892      893      894      895      896 
## 3.421102 3.147585 3.390411 3.382452 3.205523 3.382452 3.384611 3.396270 
##      897      898      899      900      901      902      903      904 
## 3.307831 3.396270 3.307831 3.286809 3.273018 3.365116 3.365116 3.414982 
##      905      906      907      908      909      910      911      912 
## 3.414982 3.249237 3.373894 3.478528 3.371070 3.374957 3.239103 3.257246 
##      913      914      915      916      917      918      919      920 
## 3.199248 3.233827 3.374957 3.288329 3.518645 3.409680 3.302814 3.299080 
##      921      922      923      924      925      926      927      928 
## 3.226959 3.302814 3.299080 3.409680 3.286809 3.296605 3.249805 3.291686 
##      929      930      931      932      933      934      935      936 
## 3.286809 3.282576 3.432759 3.373187 3.354416 3.373187 3.432759 3.278375 
##      937      938      939      940      941      942      943      944 
## 3.278375 3.093726 3.377444 3.464239 3.223149 3.211869 3.197689 3.212934 
##      945      946      947      948      949      950      951      952 
## 3.302190 3.190987 3.182844 3.304064 3.278973 3.278973 3.278973 3.304064 
##      953      954      955      956      957      958      959      960 
## 3.313529 3.193041 3.344922 3.293526 3.254946 3.226413 3.443060 3.330973 
##      961      962      963      964      965      966      967      968 
## 3.292912 3.388955 3.435120 3.279572 3.292912 3.305317 3.260134 3.285597 
##      969      970      971      972      973      974      975      976 
## 3.257822 3.422643 3.186901 3.186901 3.195102 3.290463 3.274798 3.381016 
##      977      978      979      980      981      982      983      984 
## 3.381016 3.290463 3.394800 3.093295 3.253800 3.214001 3.364768 3.253800 
##      985      986      987      988      989      990      991      992 
## 3.093295 3.376022 3.218825 3.391870 3.359913 3.226140 3.359913 3.394800 
##      993      994      995      996      997      998      999     1000 
## 3.450302 3.394800 3.204473 3.349649 3.513642 3.513642 3.243585 3.443060 
##     1001     1002     1003     1004     1005     1006     1007     1008 
## 3.358534 3.213467 3.264202 3.412703 3.312893 3.412703 3.264202 3.263036 
##     1009     1010     1011     1012     1013     1014     1015     1016 
## 3.268299 3.222606 3.269475 3.275393 3.317039 3.359913 3.324449 3.176314 
##     1017     1018     1019     1020     1021     1022     1023     1024 
## 3.266539 3.336909 3.336909 3.400705 3.129824 3.129824 3.403681 3.311623 
##     1025     1026     1027     1028     1029     1030     1031     1032 
## 3.354416 3.281373 3.338237 3.434726 3.392601 3.354416 3.397007 3.384611 
##     1033     1034     1035     1036     1037     1038     1039     1040 
## 3.312893 3.364072 3.258111 3.203948 3.357158 3.353050 3.275988 3.270654 
##     1041     1042     1043     1044     1045     1046     1047     1048 
## 3.349310 3.405175 3.270654 3.231345 3.440666 3.423414 3.348295 3.396270 
##     1049     1050     1051     1052     1053     1054     1055     1056 
## 3.239662 3.239103 3.396270 3.274204 3.514095 3.304690 3.300944 3.300944 
##     1057     1058     1059     1060     1061     1062     1063     1064 
## 3.253227 3.355785 3.194070 3.253227 3.126155 3.248102 3.319284 3.186901 
##     1065     1066     1067     1068     1069     1070     1071     1072 
## 3.305945 3.340902 3.434332 3.147585 3.147585 3.309724 3.243585 3.342239 
##     1073     1074     1075     1076     1077     1078     1079     1080 
## 3.388229 3.310989 3.342239 3.265954 3.205523 3.270064 3.270064 3.315441 
##     1081     1082     1083     1084     1085     1086     1087     1088 
## 3.187410 3.315441 3.383171 3.273018 3.383171 3.421872 3.292912 3.339568 
##     1089     1090     1091     1092     1093     1094     1095     1096 
## 3.119338 3.119338 3.204473 3.329662 3.401448 3.255520 3.417651 3.232999 
##     1097     1098     1099     1100     1101     1102     1103     1104 
## 3.417651 3.277180 3.323154 3.277180 3.297841 3.363376 3.477678 3.363376 
##     1105     1106     1107     1108     1109     1110     1111     1112 
## 3.319284 3.438282 3.317359 3.258977 3.309092 3.159096 3.318642 3.529250 
##     1113     1114     1115     1116     1117     1118     1119     1120 
## 3.330973 3.275393 3.547218 3.391870 3.391870 3.391870 3.402935 3.513642 
##     1121     1122     1123     1124     1125     1126     1127     1128 
## 3.321861 3.442261 3.469242 3.170351 3.443860 3.283783 3.505550 3.445463 
##     1129     1130     1131     1132     1133     1134     1135     1136 
## 3.206576 3.179821 3.265369 3.510031 3.367560 3.392601 3.299701 3.328355 
##     1137     1138     1139     1140     1141     1142     1143     1144 
## 3.175316 3.175316 3.370367 3.310989 3.374603 3.314165 3.414221 3.409680 
##     1145     1146     1147     1148     1149     1150     1151     1152 
## 3.385333 3.318642 3.345595 3.201332 3.332945 3.203424 3.317359 3.457638 
##     1153     1154     1155     1156     1157     1158     1159     1160 
## 3.298461 3.227505 3.435909 3.298461 3.295988 3.557887 3.408174 3.192013 
##     1161     1162     1163     1164     1165     1166     1167     1168 
## 3.167394 3.266539 3.292912 3.246686 3.246686 3.280772 3.209746 3.312893 
##     1169     1170     1171     1172     1173     1174     1175     1176 
## 3.441463 3.347619 3.253800 3.397007 3.216140 3.355785 3.355785 3.441463 
##     1177     1178     1179     1180     1181     1182     1183     1184 
## 3.418034 3.391870 3.491007 3.315441 3.315441 3.214001 3.194070 3.408927 
##     1185     1186     1187     1188     1189     1190     1191     1192 
## 3.405923 3.394066 3.416506 3.394066 3.405923 3.259266 3.244147 3.419949 
##     1193     1194     1195     1196     1197     1198     1199     1200 
## 3.387504 3.427678 3.384251 3.462582 3.326399 3.341570 3.359223 3.326399 
##     1201     1202     1203     1204     1205     1206     1207     1208 
## 3.341570 3.333604 3.282576 3.202900 3.373187 3.373187 3.373187 3.185376 
##     1209     1210     1211     1212     1213     1214     1215     1216 
## 3.373187 3.448684 3.413462 3.435120 3.413462 3.225868 3.194586 3.275393 
##     1217     1218     1219     1220     1221     1222     1223     1224 
## 3.321861 3.313529 3.313529 3.260134 3.157161 3.157161 3.317359 3.177313 
##     1225     1226     1227     1228     1229     1230     1231     1232 
## 3.078812 3.254372 3.366861 3.257822 3.584959 3.352368 3.359223 3.331958 
##     1233     1234     1235     1236     1237     1238     1239     1240 
## 3.352368 3.204473 3.384611 3.479380 3.350328 3.384611 3.318642 3.436699 
##     1241     1242     1243     1244     1245     1246     1247     1248 
## 3.353050 3.217212 3.258977 3.302814 3.497571 3.364768 3.359913 3.364768 
##     1249     1250     1251     1252     1253     1254     1255     1256 
## 3.400705 3.395535 3.395535 3.359223 3.387504 3.336247 3.336247 3.470080 
##     1257     1258     1259     1260     1261     1262     1263     1264 
## 3.353050 3.384611 3.415743 3.415743 3.252941 3.424961 3.213467 3.309092 
##     1265     1266     1267     1268     1269     1270     1271     1272 
## 3.295988 3.387504 3.387504 3.177313 3.402191 3.535804 3.598470 3.378870 
##     1273     1274     1275     1276     1277     1278     1279     1280 
## 3.504657 3.355100 3.339568 3.311940 3.289242 3.397007 3.311940 3.220983 
##     1281     1282     1283     1284     1285     1286     1287     1288 
## 3.391870 3.391870 3.329335 3.270654 3.402935 3.136769 3.396270 3.328355 
##     1289     1290     1291     1292     1293     1294     1295     1296 
## 3.381734 3.381734 3.351007 3.311940 3.498893 3.356815 3.311940 3.431975 
##     1297     1298     1299     1300     1301     1302     1303     1304 
## 3.431975 3.383890 3.510031 3.296605 3.552528 3.401448 3.260134 3.325748 
##     1305     1306     1307     1308     1309     1310     1311     1312 
## 3.333604 3.348972 3.209217 3.405923 3.209217 3.394800 3.348972 3.437490 
##     1313     1314     1315     1316     1317     1318     1319     1320 
## 3.283783 3.405923 3.405923 3.346943 3.524614 3.204473 3.346943 3.218287 
##     1321     1322     1323     1324     1325     1326     1327     1328 
## 3.209746 3.562804 3.256095 3.256670 3.418799 3.418799 3.418799 3.418799 
##     1329     1330     1331     1332     1333     1334     1335     1336 
## 3.439870 3.356471 3.356471 3.312258 3.314803 3.242742 3.372128 3.434332 
##     1337     1338     1339     1340     1341     1342     1343     1344 
## 3.492314 3.492314 3.492314 3.372481 3.372481 3.372481 3.363376 3.372481 
##     1345     1346     1347     1348     1349     1350     1351     1352 
## 3.126155 3.315441 3.475134 3.382811 3.382811 3.413462 3.254372 3.386779 
##     1353     1354     1355     1356     1357     1358     1359     1360 
## 3.355443 3.355443 3.388229 3.477678 3.475980 3.376732 3.359913 3.122736 
##     1361     1362     1363     1364     1365     1366     1367     1368 
## 3.245274 3.289852 3.122736 3.301567 3.387141 3.347619 3.366162 3.393333 
##     1369     1370     1371     1372     1373     1374     1375     1376 
## 3.302190 3.433545 3.248669 3.330973 3.248669 3.325748 3.402563 3.373187 
##     1377     1378     1379     1380     1381     1382     1383     1384 
## 3.290158 3.542434 3.388955 3.368261 3.368261 3.394800 3.319284 3.319284 
##     1385     1386     1387     1388     1389     1390     1391     1392 
## 3.374248 3.304064 3.366861 3.366861 3.407423 3.434332 3.493187 3.316719 
##     1393     1394     1395     1396     1397     1398     1399     1400 
## 3.390411 3.322507 3.441463 3.284085 3.278674 3.364072 3.394800 3.383890 
##     1401     1402     1403     1404     1405     1406     1407     1408 
## 3.320571 3.320571 3.352368 3.384611 3.314803 3.355100 3.321861 3.491442 
##     1409     1410     1411     1412     1413     1414     1415     1416 
## 3.323801 3.491442 3.406672 3.428847 3.321861 3.208687 3.198728 3.468405 
##     1417     1418     1419     1420     1421     1422     1423     1424 
## 3.198728 3.376732 3.351007 3.336909 3.351007 3.368962 3.408174 3.448684 
##     1425     1426     1427     1428     1429     1430     1431     1432 
## 3.312258 3.312258 3.353733 3.351007 3.344250 3.341570 3.371070 3.351007 
##     1433     1434     1435     1436     1437     1438     1439     1440 
## 3.506443 3.478528 3.188430 3.188430 3.206576 3.374248 3.404427 3.375312 
##     1441     1442     1443     1444     1445     1446     1447     1448 
## 3.382452 3.348633 3.407423 3.397745 3.375312 3.348633 3.407423 3.413462 
##     1449     1450     1451     1452     1453     1454     1455     1456 
## 3.411946 3.382452 3.382452 3.336247 3.434332 3.345595 3.109043 3.418799 
##     1457     1458     1459     1460     1461     1462     1463     1464 
## 3.483659 3.345595 3.302190 3.343579 3.457638 3.451518 3.413462 3.408174 
##     1465     1466     1467     1468     1469     1470     1471     1472 
## 3.411946 3.411946 3.367560 3.387504 3.367560 3.351687 3.204473 3.412703 
##     1473     1474     1475     1476     1477     1478     1479     1480 
## 3.336909 3.468405 3.199248 3.549143 3.199248 3.549143 3.372834 3.302814 
##     1481     1482     1483     1484     1485     1486     1487     1488 
## 3.514548 3.302814 3.352368 3.316079 3.391140 3.387866 3.397007 3.475980 
##     1489     1490     1491     1492     1493     1494     1495     1496 
## 3.520933 3.460929 3.388229 3.520933 3.457638 3.340234 3.466735 3.414982 
##     1497     1498     1499     1500     1501     1502     1503     1504 
## 3.340234 3.398483 3.408550 3.398483 3.356128 3.313529 3.370015 3.447877 
##     1505     1506     1507     1508     1509     1510     1511     1512 
## 3.343579 3.412703 3.399223 3.343579 3.376732 3.341570 3.452737 3.386779 
##     1513     1514     1515     1516     1517     1518     1519     1520 
## 3.431975 3.441463 3.378157 3.378157 3.477678 3.443060 3.351687 3.420333 
##     1521     1522     1523     1524     1525     1526     1527     1528 
## 3.443060 3.405175 3.477678 3.408927 3.482800 3.429627 3.422643 3.396270 
##     1529     1530     1531     1532     1533     1534     1535     1536 
## 3.328355 3.383890 3.439870 3.458048 3.384611 3.327702 3.426512 3.397745 
##     1537     1538     1539     1540     1541     1542     1543     1544 
## 3.474288 3.532049 3.455999 3.381016 3.466735 3.378870 3.405549 3.146636 
##     1545     1546     1547     1548     1549     1550     1551     1552 
## 3.295371 3.414221 3.404427 3.450302 3.140039 3.370367 3.390411 3.391140 
##     1553     1554     1555     1556     1557     1558     1559     1560 
## 3.451112 3.372128 3.413082 3.394066 3.409680 3.413082 3.384611 3.329662 
##     1561     1562     1563     1564     1565     1566     1567     1568 
## 3.329662 3.329662 3.372834 3.372834 3.372834 3.419566 3.411190 3.372834 
##     1569     1570     1571     1572     1573     1574     1575     1576 
## 3.397007 3.462582 3.427289 3.456818 3.353050 3.468405 3.475980 3.345595 
##     1577     1578     1579     1580     1581     1582     1583     1584 
## 3.312258 3.446267 3.402191 3.462582 3.368962 3.462582 3.457228 3.454366 
##     1585     1586     1587     1588     1589     1590     1591     1592 
## 3.414221 3.372481 3.359223 3.490572 3.361296 3.409303 3.450302 3.516365 
##     1593     1594     1595     1596     1597     1598     1599 
## 3.455182 3.411190 3.460104 3.487970 3.455182 3.478103 3.468405
# (observed - expected) / expected
Wine$pH.error <- (Wine$pH.predictions - Wine$pH)/Wine$pH
Wine$pH.error
##    [1] -4.058113e-02  4.011084e-02  2.257786e-02 -5.425041e-03
##    [5] -4.058113e-02 -3.978157e-02  1.098386e-02 -3.493558e-03
##    [9] -3.885919e-03 -1.046873e-04  4.325360e-02 -1.046873e-04
##   [13] -1.739337e-02  2.056747e-02  3.334473e-02  2.990070e-02
##   [17] -3.997154e-03  6.483057e-02 -3.058060e-03  9.378240e-02
##   [21] -3.503126e-02 -4.723504e-02  5.286209e-02  4.150376e-02
##   [25] -4.156535e-03  3.670114e-02  2.352380e-02  5.286209e-02
##   [29] -2.356514e-02 -1.067620e-02  1.951111e-02 -1.595351e-02
##   [33]  4.199573e-02 -1.396547e-02  5.320976e-02 -1.649575e-02
##   [37] -2.410152e-02  2.884016e-02 -1.233770e-02  1.106930e-02
##   [41]  1.106930e-02  1.467856e-03  4.713272e-02  2.755233e-03
##   [45] -1.955575e-02 -7.404256e-02  1.653892e-02  8.179636e-03
##   [49]  3.229864e-02  5.778119e-02  3.625450e-02  1.102646e-02
##   [53]  1.447661e-02  2.712793e-02  5.882376e-02  2.889011e-03
##   [57]  4.748751e-02  2.980308e-02 -2.752629e-02 -1.038048e-02
##   [61] -4.905697e-02  3.284755e-02 -5.328801e-03  1.373265e-05
##   [65] -9.860140e-03 -9.860140e-03 -1.594606e-02 -5.605342e-03
##   [69] -1.297810e-02 -4.946895e-03  7.912299e-03  4.097535e-03
##   [73]  6.932027e-03 -5.069914e-03 -1.914254e-02 -8.003675e-02
##   [77] -8.003675e-02 -3.272703e-02  7.669549e-03  4.430747e-02
##   [81]  1.541989e-02  5.822925e-02 -3.282265e-03  6.282722e-02
##   [85]  1.122010e-03 -1.853116e-03  1.211575e-01 -1.369354e-02
##   [89]  3.209545e-02 -2.406026e-02  3.045037e-02  1.211575e-01
##   [93]  1.209511e-01 -1.369354e-02 -5.812016e-02 -6.750185e-02
##   [97] -1.288442e-02  4.387723e-02 -1.585642e-02  5.746293e-03
##  [101] -3.095432e-02  7.131062e-03  5.746293e-03  7.441894e-03
##  [105]  1.489033e-02  7.441894e-03  7.623460e-02 -6.047486e-03
##  [109] -1.581042e-02  2.270337e-02  4.605310e-02  4.791882e-02
##  [113]  4.811648e-02 -3.558613e-02  4.605310e-02  5.599351e-02
##  [117] -2.645359e-02 -8.436264e-03  2.298526e-02  1.531135e-02
##  [121]  1.300255e-02  2.298526e-02 -3.296361e-02 -3.302363e-02
##  [125] -1.408888e-02  3.408605e-02 -7.212667e-02 -7.305851e-02
##  [129] -2.812830e-02  1.786190e-02  1.510424e-02 -3.029842e-02
##  [133] -3.029842e-02  1.219490e-02  2.747246e-02  3.197247e-02
##  [137]  2.090774e-02 -2.977675e-02  2.074889e-03  5.093187e-03
##  [141]  3.197247e-02  2.090774e-02 -2.638336e-02  3.869507e-02
##  [145] -2.638336e-02  3.723174e-02 -1.347066e-02  7.727583e-02
##  [149] -2.354971e-03 -6.364955e-02  1.127935e-02  1.710538e-01
##  [153]  4.616257e-03  4.616257e-03 -1.223484e-02 -1.223484e-02
##  [157] -1.223484e-02 -1.223484e-02 -2.574403e-02 -5.148337e-02
##  [161]  4.216367e-02  5.795983e-02  5.697174e-03 -4.014448e-03
##  [165] -1.759175e-03  3.872371e-02  2.041143e-02  3.908171e-03
##  [169] -1.694823e-02  1.146381e-01 -1.202217e-02  1.345392e-02
##  [173]  1.345392e-02 -1.471820e-02  1.575122e-02  1.962480e-02
##  [177]  1.575122e-02 -6.585190e-02 -2.626666e-02  2.639635e-02
##  [181]  2.639635e-02  4.105832e-02 -1.797766e-02 -3.871880e-02
##  [185] -3.155496e-02  2.187231e-04  4.726169e-03 -1.477848e-02
##  [189]  3.909328e-02  3.949960e-02  2.312394e-02 -3.398195e-02
##  [193]  3.884842e-02  2.123454e-02  2.123454e-02  2.330715e-02
##  [197] -2.832698e-02  5.194460e-03 -1.717568e-02 -3.958106e-02
##  [201]  1.991611e-03  7.509838e-02  1.727191e-02  1.531135e-02
##  [205]  1.487502e-02 -4.254025e-02 -4.254025e-02  1.165796e-02
##  [209]  3.692609e-02 -4.820515e-02 -2.942424e-02 -1.192836e-02
##  [213] -3.006351e-02  1.801379e-03  2.858515e-02  1.078491e-02
##  [217]  4.397327e-02 -1.636797e-02 -8.262934e-03  8.988611e-03
##  [221] -1.294077e-04  6.707811e-02 -1.467359e-03 -7.686920e-04
##  [225] -8.011816e-03 -6.440744e-03  6.864116e-02 -4.118758e-02
##  [229] -6.440744e-03 -1.777939e-02  7.000821e-03 -7.640895e-03
##  [233] -1.094644e-02 -1.777939e-02 -9.080617e-03  4.978960e-03
##  [237]  4.978960e-03  4.657007e-03  4.978960e-03 -9.080617e-03
##  [241]  8.450442e-02 -1.210525e-02  8.710867e-02 -2.469239e-02
##  [245] -2.469239e-02 -6.441989e-02  1.577785e-03  3.428872e-02
##  [249]  1.318231e-03 -6.441989e-02 -1.679488e-02 -2.124776e-02
##  [253] -6.921130e-03  2.615598e-02 -2.124776e-02  1.858006e-02
##  [257]  6.801263e-03 -1.808004e-02  8.368608e-02 -4.597520e-02
##  [261]  1.608158e-02  7.027054e-03 -9.253354e-04  1.836413e-02
##  [265] -1.338913e-03 -2.562978e-02 -8.063225e-02 -7.045158e-03
##  [269] -7.514813e-02 -4.395190e-02 -5.981858e-03 -4.395190e-02
##  [273] -2.115704e-02 -6.547181e-03 -8.376540e-03 -5.981858e-03
##  [277] -7.514813e-02 -4.395190e-02 -1.223708e-02 -3.252538e-02
##  [281]  6.422690e-03  2.471305e-02 -6.411843e-04 -3.252538e-02
##  [285] -2.803249e-02 -2.803249e-02 -2.041954e-04  9.457244e-04
##  [289] -1.537928e-02 -2.520680e-02 -1.537928e-02 -4.240423e-02
##  [293] -1.091029e-02  1.273490e-04  2.992977e-04  3.481848e-02
##  [297] -1.812848e-03 -7.638077e-02 -7.795902e-02 -7.569361e-02
##  [301] -3.473215e-03 -1.981479e-02 -5.982575e-02 -8.467244e-03
##  [305]  2.134346e-02  1.869071e-02 -6.674758e-03 -8.119458e-04
##  [309] -1.449480e-03 -1.182881e-02  1.869071e-02  1.804531e-02
##  [313] -1.768283e-02 -4.548054e-03 -5.583934e-03 -7.797382e-03
##  [317] -2.229909e-02 -2.394459e-02 -4.989494e-02 -2.394459e-02
##  [321] -4.989494e-02 -4.488874e-04  1.587802e-02 -3.100152e-02
##  [325]  1.591018e-02  1.591018e-02  1.921293e-02 -2.992848e-02
##  [329] -3.616076e-02 -4.273238e-03 -1.428618e-02 -1.428618e-02
##  [333]  3.003692e-02  5.601185e-02 -1.123736e-03 -2.399304e-02
##  [337] -2.665282e-02  8.129722e-03 -2.555915e-02 -7.851623e-03
##  [341] -7.171360e-03 -1.281291e-02 -4.357686e-02 -4.357686e-02
##  [345] -6.193352e-03 -5.628452e-02 -4.577536e-02 -8.061912e-05
##  [349]  3.603367e-02 -3.135146e-02 -3.522829e-02 -2.863108e-02
##  [353]  2.425393e-02 -4.871486e-02  6.901175e-02 -3.824755e-02
##  [357] -4.147703e-02 -2.316650e-02 -1.593548e-02 -3.087596e-02
##  [361]  3.438905e-02 -2.504547e-02  1.419493e-02  2.816224e-02
##  [365] -5.105495e-03 -1.895449e-02 -5.105495e-03  1.486017e-03
##  [369]  1.850286e-03  1.152368e-02 -4.835414e-02  5.293016e-03
##  [373] -1.924860e-02  1.025187e-01  2.236891e-03  3.334723e-02
##  [377] -4.175660e-02  1.152368e-02  1.052295e-03 -2.547763e-03
##  [381]  3.086987e-02 -1.109678e-02  3.086987e-02  3.086987e-02
##  [385] -2.312817e-02 -1.820402e-02 -2.240357e-02 -2.735868e-03
##  [389] -2.655918e-02  1.313343e-02 -1.901296e-02 -1.109678e-02
##  [393]  1.459528e-02  3.380770e-02  9.697451e-03 -3.921549e-02
##  [397] -1.375877e-02 -2.091316e-02 -2.091316e-02  5.362796e-02
##  [401] -1.375877e-02  2.020405e-02 -3.273054e-03 -8.871423e-03
##  [405]  2.584362e-02 -2.411509e-02 -9.501073e-03 -2.604063e-02
##  [409] -3.820067e-03  9.860802e-03 -3.512042e-02 -3.733742e-02
##  [413]  3.209694e-02 -2.051988e-02  2.365620e-03 -1.406185e-02
##  [417] -3.124911e-02 -1.271213e-02 -4.055573e-02 -5.913727e-02
##  [421] -1.633750e-02 -3.026654e-02 -1.288129e-02  1.058721e-02
##  [425] -1.288129e-02 -3.026654e-02 -1.481883e-02 -1.213415e-02
##  [429]  3.377645e-03 -1.724969e-02  1.058721e-02  2.370768e-02
##  [433]  3.370467e-02 -2.355017e-02  8.195025e-03 -2.355017e-02
##  [437]  1.525273e-02 -1.209223e-02  8.195025e-03  1.003102e-01
##  [441]  6.685381e-02  1.357895e-02 -1.427514e-03 -1.000326e-02
##  [445] -5.842234e-03 -1.789827e-04 -6.834714e-03  7.499209e-03
##  [449]  2.693836e-02  7.736240e-05  7.736240e-05  7.491583e-02
##  [453] -5.940182e-03 -2.582383e-02 -2.815786e-02 -3.182166e-02
##  [457] -3.414068e-02 -1.041498e-02 -2.582383e-02 -4.418845e-02
##  [461] -3.293685e-02  1.208303e-02 -1.019339e-02  6.479618e-02
##  [465]  4.973685e-02 -7.750685e-02  7.229181e-03 -1.663096e-02
##  [469] -1.455096e-02  2.337447e-02  2.810195e-03  5.378893e-03
##  [473] -2.179641e-02 -1.703296e-02 -6.463008e-03  2.527513e-02
##  [477] -9.378998e-03  2.330155e-03  2.527513e-02  1.478174e-02
##  [481]  1.933446e-02  2.652408e-02  4.158260e-02  4.142121e-02
##  [485]  3.206794e-02  3.415382e-03  3.415382e-03  4.291044e-03
##  [489] -5.934834e-02 -2.903514e-02 -2.450948e-02 -2.937124e-02
##  [493] -1.838050e-02 -6.167548e-02  1.935722e-04  4.743028e-03
##  [497] -2.752629e-02  1.946121e-02  4.743028e-03 -6.167548e-02
##  [501] -2.752629e-02 -1.441545e-03 -1.441545e-03  7.355896e-04
##  [505] -7.512801e-03  5.637338e-03 -1.878769e-02 -1.411951e-02
##  [509]  6.052703e-03  1.234174e-03  2.185669e-02  6.052703e-03
##  [513]  4.453483e-02  2.314692e-02  2.314692e-02 -1.250970e-02
##  [517] -5.409074e-02  2.807138e-03 -1.740302e-02  1.108302e-03
##  [521] -5.898036e-02 -4.016255e-02 -2.283603e-02  5.413617e-03
##  [525]  3.510715e-03 -1.857385e-02  1.108302e-03 -1.802238e-03
##  [529] -1.169333e-02  3.755646e-02 -3.999676e-02 -1.598526e-02
##  [533] -1.598526e-02 -4.985132e-03 -2.174163e-02 -3.999676e-02
##  [537]  3.755646e-02 -2.150386e-02 -4.125557e-02 -2.971149e-02
##  [541] -4.170618e-03  4.096157e-03  3.648307e-02  2.615647e-02
##  [545]  5.075770e-02  5.347750e-02 -2.770549e-02 -2.643506e-02
##  [549] -9.937007e-03 -7.043036e-03  2.851688e-02  1.697412e-02
##  [553]  3.746019e-02 -6.100707e-02  1.355883e-02  1.355883e-02
##  [557]  2.558157e-02  1.239523e-02  2.558157e-02 -7.303726e-02
##  [561] -2.359141e-02  6.837219e-03  5.402720e-02  1.156939e-02
##  [565] -7.303726e-02 -2.359141e-02 -1.539553e-02 -1.539553e-02
##  [569] -3.172275e-02 -2.892578e-02 -5.755888e-02 -2.892578e-02
##  [573]  1.837244e-02 -4.092711e-02 -5.538617e-03 -3.226358e-02
##  [577] -3.804373e-02 -1.102462e-02 -8.549078e-03  1.077124e-02
##  [581] -3.233238e-02 -3.233238e-02 -2.398240e-02  4.345188e-02
##  [585] -5.570151e-02 -3.448766e-02  9.451085e-03  2.938203e-02
##  [589] -4.011805e-02  4.757444e-02 -2.939892e-02  9.381519e-02
##  [593] -2.939892e-02 -2.868736e-02 -2.317189e-02 -5.041815e-03
##  [597]  1.513395e-02  2.186583e-03  2.855624e-02  2.266073e-02
##  [601]  7.387920e-03 -1.609267e-02  1.148631e-02 -1.609267e-02
##  [605]  4.328124e-02 -1.225211e-03 -3.236657e-02 -1.330692e-02
##  [609] -3.978329e-02 -2.736466e-02 -3.468250e-02  2.419973e-02
##  [613]  1.348421e-02  1.525273e-02  1.290941e-01 -8.292969e-02
##  [617] -8.292969e-02  3.294523e-02 -6.433082e-04  1.649728e-02
##  [621]  1.003493e-02  8.677233e-03  2.379314e-02 -4.594200e-02
##  [625] -1.388844e-02 -1.388844e-02 -2.798986e-02 -2.798986e-02
##  [629]  2.195793e-03  4.026006e-02  2.195793e-03 -2.066332e-02
##  [633]  1.327457e-02 -7.257185e-02  2.229615e-02  1.467369e-03
##  [637] -1.031603e-02 -8.600220e-03 -7.530182e-03  2.981078e-02
##  [641] -5.611566e-02 -6.970368e-02 -5.611566e-02 -6.970368e-02
##  [645] -5.611566e-02 -9.622311e-03 -7.886150e-03 -6.581059e-03
##  [649] -1.550301e-02  8.214107e-02  9.615415e-02 -6.226116e-02
##  [653] -9.967552e-03  3.533502e-02  6.324620e-02  2.699951e-02
##  [657]  9.615415e-02  6.008761e-02  2.098873e-02 -9.547155e-03
##  [661]  2.098873e-02  1.466285e-02  7.969969e-03  5.341341e-02
##  [665]  2.260578e-02  3.209545e-02  3.725176e-02  6.819983e-02
##  [669]  1.139172e-02  6.819983e-02  4.545220e-02  3.096265e-02
##  [673]  7.878457e-03  3.096265e-02  2.710188e-02 -5.380037e-03
##  [677]  2.710188e-02 -3.629888e-02 -5.269973e-02  1.226128e-02
##  [681] -4.921856e-03  3.687062e-02 -1.168945e-02 -2.049131e-02
##  [685] -1.878769e-02 -2.049131e-02 -1.756256e-02 -9.205729e-03
##  [689] -1.560491e-02  3.138262e-03 -8.137921e-02 -7.239879e-02
##  [693]  7.961849e-02 -1.820635e-02 -1.489032e-02 -8.261135e-02
##  [697] -2.061000e-02 -2.061000e-02 -2.367968e-02  4.067885e-04
##  [701]  2.025527e-02 -2.061000e-02 -2.039686e-02  8.930511e-03
##  [705] -1.109019e-02 -2.904139e-02 -4.526116e-03 -3.080974e-04
##  [709]  7.704952e-04 -1.482461e-02  2.684674e-02 -3.572019e-02
##  [713] -2.268373e-02 -5.624079e-03 -2.493648e-02  2.853812e-02
##  [717] -5.624079e-03  4.756452e-02  2.727673e-02 -1.969076e-02
##  [721]  2.727673e-02  5.485943e-02  4.908418e-02  1.549437e-01
##  [725] -6.043125e-02 -9.780503e-03 -3.395666e-02 -2.568701e-03
##  [729] -2.568701e-03 -5.072004e-02  1.237114e-02 -5.421126e-02
##  [733] -7.956000e-03 -4.515984e-02 -3.939414e-02  1.927307e-02
##  [737]  1.927307e-02 -5.108126e-02  5.353358e-02 -2.508833e-02
##  [741] -4.747746e-02  1.240038e-02 -5.472794e-03 -3.982671e-03
##  [745]  4.258120e-03 -4.110464e-02  3.906893e-03 -3.813128e-03
##  [749] -3.887201e-02 -4.110464e-02  4.465150e-03  4.465150e-03
##  [753] -1.009280e-02  4.465150e-03  7.131271e-02 -5.360089e-02
##  [757] -4.797518e-02  1.837659e-02  1.837659e-02  2.864354e-02
##  [761]  8.613668e-03 -4.294548e-03 -9.915335e-03 -4.294548e-03
##  [765] -1.680591e-02 -1.818574e-02  4.236254e-02 -3.026228e-02
##  [769] -4.351140e-02 -1.721838e-02 -4.351140e-02 -1.558731e-02
##  [773] -1.249306e-02  1.038285e-02  7.131062e-03 -2.023199e-02
##  [777] -4.207614e-03 -1.205212e-03 -1.608793e-02 -2.858187e-02
##  [781]  2.897978e-02 -5.949165e-02 -3.109524e-02 -5.949165e-02
##  [785] -4.330415e-02  5.823438e-04  5.823438e-04  7.695768e-03
##  [789]  7.695768e-03  6.251756e-02  9.190473e-03  1.258632e-02
##  [793] -2.469372e-02 -7.771406e-03  9.062379e-03  5.191171e-02
##  [797]  5.773332e-02  5.864405e-04 -1.682158e-02 -1.682158e-02
##  [801]  4.075433e-02  1.947732e-02  2.320890e-03  3.426147e-02
##  [805]  1.198788e-02 -4.372184e-03  1.080034e-02 -4.372184e-03
##  [809]  3.407069e-02  3.963311e-02  2.498533e-02 -9.908325e-03
##  [813] -2.316650e-02 -9.032983e-03 -4.178313e-02 -2.316650e-02
##  [817] -2.958922e-03 -3.403116e-02  2.974655e-02 -8.162449e-03
##  [821]  1.717836e-02 -2.806448e-02 -5.232629e-02 -5.232629e-02
##  [825]  4.463309e-02  1.079473e-02 -9.746831e-03  1.079473e-02
##  [829] -3.384150e-02 -2.120273e-02 -6.859593e-03 -2.120273e-02
##  [833] -5.892699e-02 -7.658344e-02 -1.724534e-02 -1.200285e-02
##  [837]  5.203285e-02  5.203285e-02 -1.031603e-02  8.334078e-03
##  [841] -2.925432e-02 -5.905187e-03 -1.968696e-02  2.594998e-02
##  [845]  1.085994e-02 -4.445446e-02 -4.445446e-02 -4.061554e-02
##  [849] -4.445446e-02 -4.156677e-02 -1.263854e-02 -1.263854e-02
##  [853]  3.243773e-02 -6.476239e-02 -6.476239e-02 -4.558733e-02
##  [857] -6.476239e-02 -2.336147e-02 -8.294136e-03  1.521968e-03
##  [861] -3.613375e-02 -6.736017e-02  3.637211e-02 -3.613375e-02
##  [865] -3.613375e-02 -3.664535e-02  1.273490e-04  2.073535e-04
##  [869]  1.521968e-03 -2.428160e-02 -1.898574e-02 -3.344306e-02
##  [873] -1.386251e-02  1.131252e-02  1.981878e-02 -1.009552e-02
##  [877]  1.742081e-03 -1.898574e-02  1.591430e-02 -4.077747e-02
##  [881]  3.222654e-02 -3.657786e-02  2.727673e-02 -4.077747e-02
##  [885]  1.591430e-02 -2.391207e-02 -2.322661e-02  4.432469e-03
##  [889] -2.594142e-03 -2.248912e-02 -1.441547e-02 -4.179842e-02
##  [893] -4.495859e-03 -4.179842e-02 -4.118663e-02 -6.938515e-03
##  [897] -6.552617e-04 -6.938515e-03 -6.552617e-04 -5.551454e-02
##  [901] -3.734754e-02 -6.002349e-02 -6.002349e-02  1.460889e-03
##  [905]  1.460889e-03  5.955692e-03 -4.750916e-03 -3.105061e-02
##  [909] -1.718062e-02  2.582280e-02  2.818361e-03  2.429111e-02
##  [913] -3.926499e-02 -1.106203e-02  2.582280e-02  5.605147e-03
##  [917] -2.799872e-02  8.781035e-03  8.527924e-04 -2.394068e-02
##  [921] -4.025039e-03  8.527924e-04 -2.394068e-02  8.781035e-03
##  [925]  3.358786e-02 -4.996976e-02 -6.345671e-02  2.865188e-02
##  [929]  3.358786e-02 -5.279849e-03 -6.971296e-02 -3.069336e-02
##  [933] -2.770549e-02 -3.069336e-02 -6.971296e-02 -3.292773e-02
##  [937] -3.292773e-02  7.728380e-03  1.121084e-02 -4.037689e-02
##  [941] -4.072956e-02  6.855587e-03 -6.928740e-03 -8.353726e-03
##  [945]  9.844103e-03  1.623776e-02 -3.257015e-02  1.351665e-02
##  [949] -3.842425e-02 -3.842425e-02 -3.842425e-02  1.351665e-02
##  [953]  1.066056e-03 -2.174821e-03 -2.763888e-02 -1.978405e-02
##  [957] -1.663265e-02 -1.932736e-02 -7.763813e-03 -8.638936e-03
##  [961]  4.206068e-02  3.637777e-02  1.630758e-02  4.113404e-02
##  [965]  4.206068e-02 -1.038410e-02  4.157626e-02  4.971143e-02
##  [969]  1.806942e-02 -1.079694e-02 -4.297274e-02 -4.297274e-02
##  [973] -7.732286e-03 -6.520940e-02 -7.636880e-03 -1.714640e-02
##  [977] -1.714640e-02  4.459140e-02  3.499991e-02 -1.172699e-02
##  [981] -1.993991e-02 -1.107676e-02  7.415447e-03 -1.993991e-02
##  [985] -1.172699e-02 -2.144301e-02  9.036135e-03 -1.399129e-02
##  [989]  6.326372e-02  2.743320e-02  6.326372e-02 -1.600008e-02
##  [993]  2.687555e-02 -1.600008e-02  1.729287e-02 -1.190286e-02
##  [997] -5.292673e-02 -5.292673e-02  3.961050e-02 -7.763813e-03
## [1001] -2.368197e-02 -1.427393e-02  1.288859e-03  1.568553e-02
## [1005]  1.935161e-02  1.568553e-02  1.288859e-03  9.314145e-04
## [1009]  1.185730e-02  1.339825e-02  5.992378e-03  3.324695e-02
## [1013]  2.694702e-02 -5.942795e-03  3.243773e-02  3.801097e-02
## [1017] -1.058280e-03  1.546399e-01  1.546399e-01 -3.662748e-02
## [1021] -2.192998e-02 -2.192998e-02  1.602414e-02  1.272877e-02
## [1025]  1.036629e-02 -1.460274e-02 -2.104486e-02 -1.583796e-02
## [1029]  6.706527e-03  1.036629e-02 -6.723101e-03  1.033170e-02
## [1033] -1.402002e-02  1.941563e-02  1.815955e-02  1.070911e-02
## [1037]  4.259567e-02  9.104604e-04  1.110741e-02 -1.782168e-02
## [1041] -6.443845e-02  7.448133e-03 -1.782168e-02 -1.782827e-02
## [1045]  2.400770e-02  3.114892e-02  1.156939e-02 -2.685665e-02
## [1049] -9.277812e-03 -9.448530e-03 -2.685665e-02  8.059550e-02
## [1053] -1.289471e-02 -2.228096e-02  5.461478e-02  5.461478e-02
## [1057]  1.346641e-02  6.532870e-02  1.275978e-03  1.346641e-02
## [1061] -4.409308e-03  1.186977e-02  5.843651e-03 -1.027926e-02
## [1065]  7.909924e-03  1.547163e-02  1.009756e-02  4.224670e-02
## [1069]  4.224670e-02 -8.340927e-05  1.999521e-02  4.444959e-02
## [1073] -3.462066e-03  2.988892e-04  4.444959e-02 -5.059474e-02
## [1077]  1.762646e-02  1.963852e-05  1.963852e-05  1.014753e-01
## [1081]  8.674017e-03  1.014753e-01 -1.076885e-02 -1.711160e-02
## [1085] -1.076885e-02  9.675379e-02 -2.576576e-02  1.506619e-02
## [1089]  3.289345e-02  3.289345e-02  7.532636e-02 -6.070921e-03
## [1093] -8.324342e-03 -2.236637e-02  2.019439e-02  2.634875e-02
## [1097]  2.019439e-02  2.411883e-02  5.496943e-02  2.411883e-02
## [1101]  6.039915e-02  3.992961e-03  8.022534e-03  3.992961e-03
## [1105]  1.197684e-02  8.293922e-03  3.344520e-02 -3.371029e-03
## [1109] -1.515111e-02 -3.439706e-03  2.112051e-02 -6.633601e-02
## [1113]  1.245385e-02  4.980540e-02  1.639492e-02  3.511828e-03
## [1117]  3.511828e-03  3.511828e-03  3.119256e-02 -1.806293e-03
## [1121]  3.808153e-02  6.571371e-04  5.577257e-03  1.613826e-02
## [1125] -4.865754e-02  4.579066e-02  3.408541e-02 -1.838650e-02
## [1129] -4.168920e-03 -9.401644e-03  2.684571e-02  5.723825e-02
## [1133] -3.680408e-03  6.706527e-03 -9.058517e-05 -1.527968e-02
## [1137] -1.387717e-02 -1.387717e-02  7.336518e-02 -8.685831e-03
## [1141] -1.038048e-02  1.350619e-02  3.461248e-02 -5.549033e-02
## [1145] -1.302247e-02 -5.987488e-02  1.999852e-02 -2.700374e-03
## [1149] -1.683038e-02 -8.227959e-03  5.260328e-03 -6.826981e-04
## [1153]  4.713034e-02 -7.723574e-04 -4.292239e-02  4.713034e-02
## [1157] -1.021390e-02  2.829115e-02 -3.457842e-03  1.013058e-02
## [1161]  8.724078e-03  1.761353e-02  4.869801e-02 -1.316521e-02
## [1165] -1.316521e-02  8.995749e-02  9.175037e-02 -2.140739e-03
## [1169]  7.210675e-02  8.318855e-03  1.681235e-02 -6.723101e-03
## [1173] -1.647101e-02 -3.011983e-02 -3.011983e-02 -4.668624e-02
## [1177] -4.524201e-02  1.249850e-02 -5.135676e-02  1.389622e-02
## [1181]  1.389622e-02  7.523680e-03 -5.500877e-02  3.614788e-02
## [1185]  1.366758e-02  1.923900e-02 -2.940183e-02  1.923900e-02
## [1189]  1.366758e-02 -1.234371e-02 -1.093065e-02 -3.934004e-02
## [1193] -3.764103e-02 -3.716901e-02  5.100953e-02 -1.631199e-02
## [1197]  3.626128e-02  5.745879e-02  6.642009e-02  3.626128e-02
## [1201]  5.745879e-02  1.945071e-02 -2.012642e-02  2.003185e-02
## [1205] -7.886150e-03 -7.886150e-03 -7.886150e-03 -4.914144e-02
## [1209] -7.886150e-03 -3.813204e-04  1.894382e-02  3.467459e-02
## [1213]  1.894382e-02  2.408496e-02  3.051160e-02  4.721731e-03
## [1217]  9.684159e-03  2.586026e-02  2.269403e-02  1.879177e-02
## [1221] -3.745082e-02 -3.745082e-02  1.759481e-02 -1.631164e-02
## [1225]  2.870484e-03  3.313410e-02  1.148546e-01 -3.724114e-03
## [1229] -2.582622e-02 -1.400934e-02 -3.197843e-03 -1.421354e-02
## [1233] -1.400934e-02 -7.903238e-03 -1.589615e-03  5.435762e-02
## [1237]  2.770792e-02 -1.589615e-03 -1.815335e-02 -3.855436e-03
## [1241]  8.162905e-02  1.810521e-02  1.210458e-02 -1.993643e-02
## [1245]  5.032165e-02 -3.588321e-02  7.689531e-02 -3.588321e-02
## [1249] -2.725806e-03 -1.313377e-03 -1.313377e-03  2.415344e-02
## [1253] -1.811491e-02 -2.733332e-02 -1.875096e-02  2.291934e-05
## [1257]  8.162905e-02  1.946121e-02 -7.051429e-03 -7.051429e-03
## [1261]  7.713285e-02 -4.597181e-02  2.014825e-02 -1.807351e-02
## [1265] -2.485570e-02  2.219992e-03  2.219992e-03  2.493981e-02
## [1269] -1.954150e-02  1.022964e-02 -2.744049e-02 -3.342095e-04
## [1273]  1.330706e-03  8.229043e-02  3.073080e-02  2.220378e-02
## [1277]  2.817821e-03  1.403194e-02  2.220378e-02  2.579072e-02
## [1281]  6.489608e-03  6.489608e-03 -1.996398e-04  3.268043e-03
## [1285]  3.815764e-03 -4.200432e-03  1.381202e-02  3.365052e-02
## [1289]  2.476773e-02  2.476773e-02 -2.676444e-03  6.668765e-03
## [1293] -3.345496e-02  5.034304e-03  6.668765e-03  7.249204e-02
## [1297]  7.249204e-02  1.011654e-02  1.740029e-02 -5.811288e-02
## [1301] -6.017785e-02 -5.515347e-02 -1.208070e-02 -1.019398e-02
## [1305]  6.846276e-02  1.483988e-02 -2.455425e-02 -1.195579e-03
## [1309] -2.455425e-02  3.499991e-02  1.483988e-02  3.227927e-02
## [1313] -1.683153e-02  1.742081e-03  1.742081e-03  2.666973e-02
## [1317] -1.210438e-01  7.695768e-03  2.666973e-02  1.097542e-01
## [1321] -1.541531e-02 -1.115202e-01  2.392914e-02  2.411003e-02
## [1325]  8.495330e-03  8.495330e-03  8.495330e-03  8.495330e-03
## [1329]  2.990127e-02  2.020405e-02  2.020405e-02  1.602993e-02
## [1333]  7.275164e-02  1.973012e-02 -5.272038e-03  1.607447e-02
## [1337]  3.937912e-02  3.937912e-02  3.937912e-02  3.714485e-03
## [1341]  3.714485e-03  3.714485e-03 -1.965454e-03  3.714485e-03
## [1345]  1.170057e-02  2.013558e-02 -1.274612e-02  1.281765e-02
## [1349]  1.281765e-02  1.015186e-03 -2.563700e-02  2.011419e-02
## [1353] -4.319656e-03 -4.319656e-03  1.141163e-02  3.502314e-02
## [1357]  3.451798e-02 -6.201876e-02  2.436383e-02  1.717793e-02
## [1361]  1.627931e-03 -2.087728e-02  1.717793e-02  3.515776e-03
## [1365] -2.105745e-02  5.271025e-02  3.893895e-02  4.410245e-02
## [1369]  9.844103e-03  3.109456e-02  8.288971e-02  3.768635e-02
## [1373]  8.288971e-02  2.646550e-02  2.486840e-02  3.471997e-02
## [1377]  5.792847e-02 -4.516597e-02  3.007760e-02  2.458492e-03
## [1381]  2.458492e-03  4.378614e-03  5.843651e-03  5.843651e-03
## [1385]  5.775811e-02  3.575683e-02  2.026084e-02  2.026084e-02
## [1389]  1.714114e-02  1.078489e-01 -2.424953e-02 -1.581042e-02
## [1393]  6.953021e-02  6.820280e-03  3.658514e-02 -7.829420e-03
## [1397] -9.464031e-03  2.563158e-02 -3.556826e-02 -1.055836e-02
## [1401]  2.171415e-02  2.171415e-02 -2.547434e-02  4.786725e-02
## [1405] -3.358525e-02  2.289644e-02  3.163381e-02 -1.371685e-02
## [1409]  1.645297e-02 -1.371685e-02 -2.666501e-02 -3.684079e-02
## [1413]  3.163381e-02  5.548929e-02 -3.975809e-04 -2.572899e-02
## [1417] -3.975809e-04  7.979839e-03  6.044530e-02  2.046158e-02
## [1421]  6.044530e-02  8.671171e-03  5.361115e-03 -1.466159e-02
## [1425]  1.602993e-02  1.602993e-02 -1.650069e-02  1.545671e-02
## [1429]  1.034749e-02  3.474412e-03  3.176453e-04  2.791630e-02
## [1433]  7.598561e-03  1.711354e-02  2.650787e-03  2.650787e-03
## [1437]  1.796066e-02 -4.412227e-02 -1.634226e-03  1.666014e-02
## [1441]  4.719863e-02  5.969400e-02  2.183182e-03  5.249902e-03
## [1445]  1.666014e-02  5.969400e-02  2.183182e-03  3.959348e-03
## [1449]  3.513530e-03  4.719863e-02  4.719863e-02  1.875893e-03
## [1453]  2.517365e-02 -1.888705e-02 -3.145083e-02 -2.320024e-02
## [1457]  5.565415e-02 -1.888705e-02 -2.590259e-02  7.102078e-03
## [1461]  1.397005e-02 -3.857439e-02  1.591125e-02  2.041143e-02
## [1465]  5.706747e-04  5.706747e-04  2.047279e-02 -3.764103e-02
## [1469]  2.047279e-02  1.259435e-02  1.126641e-01 -5.042745e-03
## [1473] -1.274870e-02  2.615526e-02  1.242013e-02  2.582832e-03
## [1477]  1.242013e-02  2.582832e-03 -7.990067e-03 -2.571852e-02
## [1481] -3.974094e-02 -2.571852e-02  6.717195e-03 -1.012555e-02
## [1485] -3.933714e-02  5.301530e-03  5.031655e-03 -1.250556e-02
## [1489] -5.351255e-02  6.084022e-03 -5.224264e-04 -5.351255e-02
## [1493] -3.147399e-02  2.461176e-02  1.366519e-02  1.939750e-02
## [1497]  2.461176e-02 -1.493238e-02 -3.440503e-02 -1.493238e-02
## [1501]  2.948719e-02  4.858501e-02  1.813145e-02 -1.769873e-02
## [1505] -4.887233e-03 -3.867510e-02 -3.156043e-02 -4.887233e-03
## [1509] -9.667280e-04  1.877128e-02 -4.975026e-03 -1.547119e-02
## [1513] -2.776925e-02  3.347227e-02 -4.301506e-02 -4.301506e-02
## [1517]  1.686484e-02  1.865668e-02  9.544366e-03 -5.717031e-03
## [1521]  1.865668e-02  4.476310e-03  1.686484e-02  2.063669e-02
## [1525]  2.737475e-02  2.991805e-02  3.716442e-02  3.230100e-02
## [1529]  5.545221e-03 -1.055836e-02  5.810010e-03 -3.943105e-02
## [1533]  5.439601e-02 -1.837701e-02  1.904166e-03  1.123353e-02
## [1537] -3.491995e-02  9.156969e-03 -3.193294e-02 -2.282764e-02
## [1541] -2.619804e-02 -6.214597e-03  3.198447e-02 -3.180424e-02
## [1545]  3.954927e-02  1.613726e-02  1.322241e-02 -2.808398e-02
## [1549]  1.291574e-02  4.023663e-02 -1.727224e-02 -1.420352e-02
## [1553] -1.677712e-02 -1.110622e-02 -3.585241e-02 -1.335295e-02
## [1557] -3.408501e-02 -3.585241e-02  5.112149e-02  3.727801e-02
## [1561]  3.727801e-02  3.727801e-02  2.517744e-02  2.517744e-02
## [1565]  2.517744e-02  8.721467e-03  2.131426e-02  2.517744e-02
## [1569]  1.706796e-02 -5.005226e-03  1.699981e-02  4.888968e-03
## [1573]  6.921935e-03 -3.117185e-02  2.536296e-02  2.625618e-02
## [1577]  3.714413e-03 -2.647826e-02 -5.207314e-03 -2.186954e-02
## [1581]  2.667176e-03 -2.186954e-02 -3.158885e-02  3.734705e-02
## [1585]  3.775720e-02  2.196384e-02  5.755470e-03 -1.674022e-02
## [1589]  2.791928e-02  3.626234e-02  3.924755e-02 -4.186235e-02
## [1593]  1.028712e-02 -2.576133e-03  2.928826e-03 -9.099503e-03
## [1597]  1.028712e-02 -2.574149e-02  2.312825e-02
ggplot(data = Wine, aes(x = quality, y = pH.error)) +
  geom_boxplot(outlier.colour = 'red')

# We can also add something interesting to our model, to check its accuracy.
# The RMS Error.

rmse <- function(error)
{
  sqrt(mean(error^2))
}

rmse(m$residuals)
## [1] 0.1095431
#Now, we train a Support Vector Machine.

library(e1071)

SVM <- svm(I(pH) ~ I(log10(TAC.acidity)), data = Wine)
Wine$pH.Predict.SVM <- predict(SVM,Wine)
predict(SVM,Wine)
##        1        2        3        4        5        6        7        8 
## 3.359582 3.296984 3.303653 3.153620 3.359582 3.364450 3.307306 3.377503 
##        9       10       11       12       13       14       15       16 
## 3.323787 3.328300 3.429305 3.328300 3.584627 3.295454 3.269955 3.269833 
##       17       18       19       20       21       22       23       24 
## 3.272676 3.281461 3.363237 3.293265 3.270937 3.335288 3.309208 3.276241 
##       25       26       27       28       29       30       31       32 
## 3.424096 3.468723 3.341267 3.309208 3.392893 3.318879 3.423758 3.413508 
##       33       34       35       36       37       38       39       40 
## 3.276872 3.410240 3.674093 3.318879 3.309208 3.291203 3.461291 3.358360 
##       41       42       43       44       45       46       47       48 
## 3.358360 3.269833 3.348562 3.279879 3.420644 3.683489 3.277142 3.271561 
##       49       50       51       52       53       54       55       56 
## 3.451458 3.569795 3.269706 3.441976 3.443327 3.272676 3.340062 3.328300 
##       57       58       59       60       61       62       63       64 
## 3.190797 3.341267 3.306373 3.371647 3.270937 3.284219 3.349785 3.395450 
##       65       66       67       68       69       70       71       72 
## 3.374594 3.374594 3.355912 3.428351 3.256122 3.291538 3.322680 3.303653 
##       73       74       75       76       77       78       79       80 
## 3.302772 3.273776 3.234591 3.267529 3.267529 3.413508 3.414691 3.275670 
##       81       82       83       84       85       86       87       88 
## 3.468723 3.281888 3.327160 3.343688 3.447592 3.412302 3.272445 3.318345 
##       89       90       91       92       93       94       95       96 
## 3.258777 3.403653 3.296984 3.272445 3.272373 3.318345 3.619731 3.695902 
##       97       98       99      100      101      102      103      104 
## 3.414300 3.398921 3.307306 3.287171 3.274128 3.303653 3.287171 3.283485 
##      105      106      107      108      109      110      111      112 
## 3.379800 3.283485 3.283726 3.443327 3.285245 3.272204 3.308251 3.276005 
##      113      114      115      116      117      118      119      120 
## 3.276241 3.214138 3.308251 3.259602 3.275779 3.315200 3.272304 3.396954 
##      121      122      123      124      125      126      127      128 
## 3.317286 3.272304 3.372240 3.294710 3.316238 3.270396 3.271435 3.272045 
##      129      130      131      132      133      134      135      136 
## 3.291876 3.499353 3.272986 3.590358 3.590358 3.445415 3.279514 3.273523 
##      137      138      139      140      141      142      143      144 
## 3.274872 3.374594 3.308251 3.308251 3.273523 3.274872 3.694897 3.478528 
##      145      146      147      148      149      150      151      152 
## 3.694897 3.272847 3.522855 3.329448 3.420644 3.277863 3.359582 3.223451 
##      153      154      155      156      157      158      159      160 
## 3.355912 3.355912 3.377503 3.377503 3.377503 3.377503 3.395954 3.413905 
##      161      162      163      164      165      166      167      168 
## 3.305453 3.335288 3.327160 3.340062 3.353460 3.282781 3.416992 3.385409 
##      169      170      171      172      173      174      175      176 
## 3.419931 3.318879 3.287753 3.304547 3.304547 3.363237 3.384303 3.424096 
##      177      178      179      180      181      182      183      184 
## 3.384303 3.319415 3.393409 3.271303 3.271303 3.263584 3.377503 3.411485 
##      185      186      187      188      189      190      191      192 
## 3.417740 3.268829 3.338862 3.312653 3.293265 3.294710 3.277567 3.452271 
##      193      194      195      196      197      198      199      200 
## 3.416235 3.328300 3.328300 3.293981 3.349785 3.146167 3.561911 3.365660 
##      201      202      203      204      205      206      207      208 
## 3.245446 3.270495 3.426087 3.396954 3.394944 3.103621 3.103621 3.296984 
##      209      210      211      212      213      214      215      216 
## 3.311159 3.158090 3.215572 3.280449 3.141258 3.278171 3.302336 3.374008 
##      217      218      219      220      221      222      223      224 
## 3.271713 3.277714 3.349785 3.298569 3.312152 3.348562 3.423420 3.271653 
##      225      226      227      228      229      230      231      232 
## 3.272270 3.326027 3.263899 3.264800 3.326027 3.406361 3.680688 3.319415 
##      233      234      235      236      237      238      239      240 
## 3.277006 3.406361 3.273134 3.390802 3.390802 3.389208 3.390802 3.273134 
##      241      242      243      244      245      246      247      248 
## 3.266274 3.136056 3.326027 2.991567 2.991567 3.376345 3.388672 3.280256 
##      249      250      251      252      253      254      255      256 
## 3.336474 3.376345 3.170355 3.403653 3.156590 3.283485 3.403653 3.288652 
##      257      258      259      260      261      262      263      264 
## 3.259602 3.433377 3.284725 3.219161 3.317286 3.369863 3.308251 3.313156 
##      265      266      267      268      269      270      271      272 
## 3.116999 3.143030 3.280256 3.294710 3.421349 3.150183 3.312653 3.150183 
##      273      274      275      276      277      278      279      280 
## 3.159039 3.272892 3.331765 3.312653 3.421349 3.150183 3.198695 3.270770 
##      281      282      283      284      285      286      287      288 
## 3.152170 3.299383 3.342476 3.270770 3.234591 3.234591 3.134667 3.369266 
##      289      290      291      292      293      294      295      296 
## 3.272939 3.143552 3.272939 3.165180 3.191436 3.419209 3.085832 3.162123 
##      297      298      299      300      301      302      303      304 
## 3.160708 3.394944 3.386506 3.409820 3.360803 3.162123 3.275670 3.348562 
##      305      306      307      308      309      310      311      312 
## 3.270495 3.183461 3.309208 3.194684 3.192078 3.363237 3.183461 3.297770 
##      313      314      315      316      317      318      319      320 
## 3.268989 3.272445 3.365660 3.399889 3.245446 3.239219 3.214138 3.239219 
##      321      322      323      324      325      326      327      328 
## 3.214138 3.257037 3.316238 3.183461 3.225581 3.225581 3.139193 3.187666 
##      329      330      331      332      333      334      335      336 
## 3.077862 3.170819 3.190163 3.190163 3.285245 3.277863 3.307306 3.131474 
##      337      338      339      340      341      342      343      344 
## 3.268829 3.308251 3.120186 3.125572 3.133691 3.173215 3.162123 3.162123 
##      345      346      347      348      349      350      351      352 
## 3.135372 3.405019 3.424431 3.033454 3.240503 3.266755 3.168993 3.266518 
##      353      354      355      356      357      358      359      360 
## 3.327729 3.042867 3.486870 3.422737 3.145637 3.169442 3.134904 3.113150 
##      361      362      363      364      365      366      367      368 
## 3.275562 3.272519 3.131061 3.115322 3.090831 3.209138 3.090831 3.169217 
##      369      370      371      372      373      374      375      376 
## 3.192724 3.255653 3.405019 3.309208 3.267317 3.351009 3.028138 3.139962 
##      377      378      379      380      381      382      383      384 
## 3.145287 3.255653 3.141167 3.276241 3.279879 3.044555 3.279879 3.279879 
##      385      386      387      388      389      390      391      392 
## 3.314172 3.359582 3.303653 3.276005 3.311159 3.251114 3.522855 3.044555 
##      393      394      395      396      397      398      399      400 
## 3.245446 3.270239 3.097835 3.131337 3.429622 3.141258 3.141258 3.270129 
##      401      402      403      404      405      406      407      408 
## 3.429622 3.340062 3.128747 3.144591 3.319415 3.271803 3.224163 3.133440 
##      409      410      411      412      413      414      415      416 
## 3.182878 3.120948 3.268989 3.266397 3.372240 3.215572 3.270396 3.271199 
##      417      418      419      420      421      422      423      424 
## 3.163613 3.403653 3.139193 3.414691 3.245446 3.422047 3.285245 3.189532 
##      425      426      427      428      429      430      431      432 
## 3.285245 3.422047 3.442648 3.238570 3.265085 3.079112 3.189532 3.295454 
##      433      434      435      436      437      438      439      440 
## 3.135602 3.125572 3.180594 3.125572 3.280256 3.148378 3.180594 3.393923 
##      441      442      443      444      445      446      447      448 
## 3.113591 3.135830 2.965397 3.208428 3.656249 3.248640 3.119799 3.261158 
##      449      450      451      452      453      454      455      456 
## 3.272596 3.135139 3.135139 3.273034 3.427387 3.180594 3.409820 3.142856 
##      457      458      459      460      461      462      463      464 
## 3.266871 3.262244 3.180594 3.138195 3.257922 3.275457 3.156301 3.271863 
##      465      466      467      468      469      470      471      472 
## 3.146976 3.225581 3.188906 3.269833 3.145287 3.271370 3.096744 3.234591 
##      473      474      475      476      477      478      479      480 
## 3.122066 3.217726 3.184640 3.237259 3.264207 3.194684 3.237259 3.255888 
##      481      482      483      484      485      486      487      488 
## 3.185837 3.252690 3.170819 3.170355 3.163613 3.186442 3.186442 3.189847 
##      489      490      491      492      493      494      495      496 
## 3.144938 3.260396 3.248073 3.259602 3.268310 3.269955 3.444710 3.169442 
##      497      498      499      500      501      502      503      504 
## 3.306373 3.387593 3.169442 3.269955 3.306373 3.169896 3.169896 3.188284 
##      505      506      507      508      509      510      511      512 
## 3.192724 3.182878 3.196675 3.145813 3.210561 3.074706 3.125904 3.210561 
##      513      514      515      516      517      518      519      520 
## 3.169442 3.166406 3.166406 3.270544 3.115322 3.173215 3.168550 3.352848 
##      521      522      523      524      525      526      527      528 
## 3.235932 3.313156 3.276741 3.256122 3.259193 3.185236 3.352848 3.386506 
##      529      530      531      532      533      534      535      536 
## 3.276005 3.219878 3.271234 3.139580 3.139580 3.217008 3.223451 3.271234 
##      537      538      539      540      541      542      543      544 
## 3.219878 3.274872 3.108551 3.145461 3.262591 3.241770 3.252946 3.152170 
##      545      546      547      548      549      550      551      552 
## 3.008427 3.261528 3.336474 3.178387 3.128138 3.262930 3.431819 3.261158 
##      553      554      555      556      557      558      559      560 
## 3.255653 3.569795 2.962596 2.962596 3.156882 2.962436 3.156882 3.097291 
##      561      562      563      564      565      566      567      568 
## 3.106127 3.265632 3.262591 3.326027 3.097291 3.106127 3.270591 3.270591 
##      569      570      571      572      573      574      575      576 
## 3.218443 3.474679 3.147251 3.474679 3.208428 3.169442 3.176783 3.133187 
##      577      578      579      580      581      582      583      584 
## 3.229096 3.269573 3.269144 3.178387 3.126558 3.126558 3.141258 3.139580 
##      585      586      587      588      589      590      591      592 
## 3.142329 3.329448 3.157478 3.338862 3.687166 3.204903 3.265362 3.421349 
##      593      594      595      596      597      598      599      600 
## 3.265362 3.205604 3.289268 3.266274 3.125904 3.133187 3.273405 3.108551 
##      601      602      603      604      605      606      607      608 
## 3.272338 3.084630 3.309691 3.084630 3.278819 3.278171 3.251114 3.270072 
##      609      610      611      612      613      614      615      616 
## 3.195344 3.468723 3.271091 3.087617 3.354686 3.280256 3.258566 3.224873 
##      617      618      619      620      621      622      623      624 
## 3.224873 3.147621 3.146885 3.152641 3.276741 3.279514 3.217726 3.298569 
##      625      626      627      628      629      630      631      632 
## 3.420644 3.420644 3.270072 3.270072 3.271622 3.311654 3.271622 3.188906 
##      633      634      635      636      637      638      639      640 
## 3.353460 3.186746 3.317286 3.271370 3.220594 3.228047 3.292913 3.271370 
##      641      642      643      644      645      646      647      648 
## 3.211275 3.236598 3.211275 3.236598 3.211275 3.309208 3.369266 3.275059 
##      649      650      651      652      653      654      655      656 
## 3.271743 3.427387 3.172242 3.208428 2.963470 3.249479 3.271370 3.243632 
##      657      658      659      660      661      662      663      664 
## 3.172242 3.132411 3.394944 3.376345 3.394944 3.343688 3.390802 3.214855 
##      665      666      667      668      669      670      671      672 
## 3.134184 3.258777 3.275155 3.152404 3.192724 3.152404 3.416992 3.275354 
##      673      674      675      676      677      678      679      680 
## 3.179481 3.275354 3.168111 3.260002 3.168111 3.271561 3.274607 3.172726 
##      681      682      683      684      685      686      687      688 
## 3.076605 3.296984 3.273348 3.276005 3.196675 3.276005 3.379800 3.264507 
##      689      690      691      692      693      694      695      696 
## 3.323787 3.280645 3.304998 3.247211 3.270937 3.268489 3.268829 3.692566 
##      697      698      699      700      701      702      703      704 
## 3.406361 3.406361 3.250846 3.140338 3.153371 3.406361 3.407242 3.328300 
##      705      706      707      708      709      710      711      712 
## 3.266274 3.271053 3.387593 3.362021 3.316761 3.239863 3.153247 3.268401 
##      713      714      715      716      717      718      719      720 
## 3.272304 3.289268 3.209138 3.386506 3.289268 3.351009 3.279161 3.397943 
##      721      722      723      724      725      726      727      728 
## 3.279161 3.270770 3.359582 3.402728 3.294344 3.267933 3.282781 3.454801 
##      729      730      731      732      733      734      735      736 
## 3.454801 3.433377 3.224163 3.266274 3.351622 3.394944 3.317286 3.290872 
##      737      738      739      740      741      742      743      744 
## 3.290872 3.289268 3.270072 3.270072 3.272596 3.264507 3.445062 3.142856 
##      745      746      747      748      749      750      751      752 
## 3.153873 3.372830 3.282328 3.272760 3.385409 3.372830 3.277567 3.277567 
##      753      754      755      756      757      758      759      760 
## 3.338862 3.277567 3.280645 3.289899 3.433688 3.280256 3.280256 3.272045 
##      761      762      763      764      765      766      767      768 
## 3.267933 3.254936 3.271622 3.254936 3.266638 3.264507 3.271164 3.321582 
##      769      770      771      772      773      774      775      776 
## 3.402728 3.301050 3.402728 3.248073 3.248357 3.304547 3.303653 3.347340 
##      777      778      779      780      781      782      783      784 
## 3.389208 3.403192 3.278171 3.408112 3.414691 3.446131 3.267097 3.446131 
##      785      786      787      788      789      790      791      792 
## 3.403653 3.227697 3.227697 3.217726 3.217726 3.272237 3.347340 3.270855 
##      793      794      795      796      797      798      799      800 
## 3.400846 3.335288 3.209849 3.154917 3.271803 3.259602 3.253706 3.253706 
##      801      802      803      804      805      806      807      808 
## 3.384303 3.273711 3.684666 3.330603 3.275354 3.284219 3.277863 3.284219 
##      809      810      811      812      813      814      815      816 
## 3.365660 3.324903 3.384303 3.097835 3.169442 3.417740 3.116999 3.169442 
##      817      818      819      820      821      822      823      824 
## 3.238570 3.172726 3.392374 3.266150 3.405019 3.688749 3.428670 3.428670 
##      825      826      827      828      829      830      831      832 
## 3.387593 3.403653 3.358360 3.403653 3.317286 3.506173 3.340664 3.506173 
##      833      834      835      836      837      838      839      840 
## 3.186442 3.144244 3.269365 3.327729 3.435558 3.435558 3.220594 3.515220 
##      841      842      843      844      845      846      847      848 
## 3.154917 3.435558 3.170819 3.355299 3.231178 3.436183 3.436183 3.342476 
##      849      850      851      852      853      854      855      856 
## 3.436183 3.436811 3.257037 3.257037 3.292564 3.261890 3.261890 3.328873 
##      857      858      859      860      861      862      863      864 
## 3.261890 3.288652 3.145287 3.413905 3.385409 3.422047 3.342476 3.385409 
##      865      866      867      868      869      870      871      872 
## 3.385409 3.382630 3.419209 3.408972 3.413905 3.340062 3.321037 3.420644 
##      873      874      875      876      877      878      879      880 
## 3.384303 3.270184 3.187666 3.271561 3.414691 3.321037 3.270937 3.389742 
##      881      882      883      884      885      886      887      888 
## 3.265362 3.333520 3.279161 3.389742 3.270937 3.268662 3.265895 3.168550 
##      889      890      891      892      893      894      895      896 
## 3.428670 3.156301 3.395954 3.384303 3.219161 3.384303 3.387593 3.403653 
##      897      898      899      900      901      902      903      904 
## 3.279161 3.403653 3.279161 3.272676 3.271164 3.355299 3.355299 3.423420 
##      905      906      907      908      909      910      911      912 
## 3.423420 3.264207 3.370459 3.494490 3.365660 3.372240 3.257483 3.267734 
##      913      914      915      916      917      918      919      920 
## 3.210561 3.252946 3.372240 3.272892 3.586919 3.418479 3.276741 3.275354 
##      921      922      923      924      925      926      927      928 
## 3.246040 3.276741 3.275354 3.418479 3.272676 3.274607 3.264507 3.273464 
##      929      930      931      932      933      934      935      936 
## 3.272676 3.272172 3.438076 3.369266 3.336474 3.369266 3.438076 3.271743 
##      937      938      939      940      941      942      943      944 
## 3.271743 3.131337 3.376345 3.471038 3.241770 3.227697 3.208428 3.229096 
##      945      946      947      948      949      950      951      952 
## 3.276486 3.199374 3.188906 3.277281 3.271803 3.271803 3.271803 3.277281 
##      953      954      955      956      957      958      959      960 
## 3.282781 3.202119 3.320494 3.273843 3.266871 3.245446 3.446856 3.300210 
##      961      962      963      964      965      966      967      968 
## 3.273711 3.393923 3.440002 3.271863 3.273711 3.277863 3.268662 3.272519 
##      969      970      971      972      973      974      975      976 
## 3.267933 3.429937 3.194027 3.194027 3.204903 3.273239 3.271370 3.382068 
##      977      978      979      980      981      982      983      984 
## 3.382068 3.273239 3.401793 3.131061 3.266397 3.230487 3.354686 3.266397 
##      985      986      987      988      989      990      991      992 
## 3.131061 3.374008 3.236598 3.397943 3.346120 3.245147 3.346120 3.401793 
##      993      994      995      996      997      998      999     1000 
## 3.453942 3.401793 3.217726 3.328300 3.574339 3.574339 3.260781 3.446856 
##     1001     1002     1003     1004     1005     1006     1007     1008 
## 3.343688 3.229793 3.269706 3.421349 3.282328 3.421349 3.269706 3.269436 
##     1009     1010     1011     1012     1013     1014     1015     1016 
## 3.270495 3.241139 3.270682 3.271435 3.285509 3.346120 3.292564 3.181158 
##     1017     1018     1019     1020     1021     1022     1023     1024 
## 3.270184 3.308251 3.308251 3.408972 3.147621 3.147621 3.412302 3.281461 
##     1025     1026     1027     1028     1029     1030     1031     1032 
## 3.336474 3.272045 3.310177 3.439678 3.398921 3.336474 3.404567 3.387593 
##     1033     1034     1035     1036     1037     1038     1039     1040 
## 3.282328 3.353460 3.268030 3.217008 3.341267 3.334108 3.271498 3.270855 
##     1041     1042     1043     1044     1045     1046     1047     1048 
## 3.327729 3.413905 3.270855 3.250576 3.444710 3.430567 3.326027 3.403653 
##     1049     1050     1051     1052     1053     1054     1055     1056 
## 3.257922 3.257483 3.403653 3.271303 3.575478 3.277567 3.276005 3.276005 
##     1057     1058     1059     1060     1061     1062     1063     1064 
## 3.266150 3.338862 3.203507 3.266150 3.146167 3.263584 3.287460 3.194027 
##     1065     1066     1067     1068     1069     1070     1071     1072 
## 3.278171 3.314172 3.439355 3.156301 3.156301 3.280256 3.260781 3.316238 
##     1073     1074     1075     1076     1077     1078     1079     1080 
## 3.392893 3.281047 3.316238 3.270072 3.219161 3.270770 3.270770 3.284219 
##     1081     1082     1083     1084     1085     1086     1087     1088 
## 3.194684 3.284219 3.385409 3.271164 3.385409 3.429305 3.273711 3.312152 
##     1089     1090     1091     1092     1093     1094     1095     1096 
## 3.143552 3.143552 3.217726 3.298569 3.409820 3.267097 3.425759 3.252171 
##     1097     1098     1099     1100     1101     1102     1103     1104 
## 3.425759 3.271622 3.291203 3.271622 3.274964 3.352235 3.492917 3.352235 
##     1105     1106     1107     1108     1109     1110     1111     1112 
## 3.287460 3.442648 3.285778 3.268310 3.279879 3.164387 3.286886 3.613075 
##     1113     1114     1115     1116     1117     1118     1119     1120 
## 3.300210 3.271435 3.652577 3.397943 3.397943 3.397943 3.411485 3.574339 
##     1121     1122     1123     1124     1125     1126     1127     1128 
## 3.289899 3.446131 3.478528 3.174714 3.447592 3.272304 3.554143 3.449100 
##     1129     1130     1131     1132     1133     1134     1135     1136 
## 3.220594 3.185236 3.269955 3.565277 3.359582 3.398921 3.275562 3.296984 
##     1137     1138     1139     1140     1141     1142     1143     1144 
## 3.180035 3.180035 3.364450 3.281047 3.371647 3.283247 3.422737 3.418479 
##     1145     1146     1147     1148     1149     1150     1151     1152 
## 3.388672 3.286886 3.321582 3.213421 3.302772 3.216290 3.285778 3.462291 
##     1153     1154     1155     1156     1157     1158     1159     1160 
## 3.275155 3.246628 3.440654 3.275155 3.274440 3.671173 3.416992 3.200741 
##     1161     1162     1163     1164     1165     1166     1167     1168 
## 3.171762 3.270184 3.273711 3.262761 3.262761 3.271984 3.224873 3.282328 
##     1169     1170     1171     1172     1173     1174     1175     1176 
## 3.445415 3.324903 3.266397 3.404567 3.233235 3.338862 3.338862 3.445415 
##     1177     1178     1179     1180     1181     1182     1183     1184 
## 3.426087 3.397943 3.519952 3.284219 3.284219 3.230487 3.203507 3.417740 
##     1185     1186     1187     1188     1189     1190     1191     1192 
## 3.414691 3.400846 3.424765 3.400846 3.414691 3.268401 3.261158 3.427710 
##     1193     1194     1195     1196     1197     1198     1199     1200 
## 3.391852 3.433999 3.387051 3.468723 3.294710 3.315200 3.344902 3.294710 
##     1201     1202     1203     1204     1205     1206     1207     1208 
## 3.315200 3.303653 3.272172 3.215572 3.369266 3.369266 3.369266 3.192078 
##     1209     1210     1211     1212     1213     1214     1215     1216 
## 3.369266 3.452271 3.422047 3.440002 3.422047 3.244847 3.204204 3.271435 
##     1217     1218     1219     1220     1221     1222     1223     1224 
## 3.289899 3.282781 3.282781 3.268662 3.162859 3.162859 3.285778 3.182300 
##     1225     1226     1227     1228     1229     1230     1231     1232 
## 3.119799 3.266638 3.358360 3.267933 3.695457 3.332933 3.344902 3.301476 
##     1233     1234     1235     1236     1237     1238     1239     1240 
## 3.332933 3.217726 3.387593 3.496087 3.329448 3.387593 3.286886 3.441312 
##     1241     1242     1243     1244     1245     1246     1247     1248 
## 3.334108 3.234591 3.268310 3.276741 3.534912 3.354686 3.346120 3.354686 
##     1249     1250     1251     1252     1253     1254     1255     1256 
## 3.408972 3.402728 3.402728 3.344902 3.391852 3.307306 3.307306 3.479858 
##     1257     1258     1259     1260     1261     1262     1263     1264 
## 3.334108 3.387593 3.424096 3.424096 3.266023 3.431819 3.229793 3.279879 
##     1265     1266     1267     1268     1269     1270     1271     1272 
## 3.274440 3.391852 3.391852 3.182300 3.410658 3.628415 3.693693 3.378655 
##     1273     1274     1275     1276     1277     1278     1279     1280 
## 3.551949 3.337666 3.312152 3.281673 3.273034 3.404567 3.281673 3.239219 
##     1281     1282     1283     1284     1285     1286     1287     1288 
## 3.397943 3.397943 3.298168 3.270855 3.411485 3.150607 3.403653 3.296984 
##     1289     1290     1291     1292     1293     1294     1295     1296 
## 3.383190 3.383190 3.330603 3.281673 3.538029 3.340664 3.281673 3.437442 
##     1297     1298     1299     1300     1301     1302     1303     1304 
## 3.437442 3.386506 3.565277 3.274607 3.662379 3.409820 3.268662 3.293981 
##     1305     1306     1307     1308     1309     1310     1311     1312 
## 3.303653 3.327160 3.224163 3.414691 3.224163 3.401793 3.327160 3.441976 
##     1313     1314     1315     1316     1317     1318     1319     1320 
## 3.272304 3.414691 3.414691 3.323787 3.601787 3.217726 3.323787 3.235932 
##     1321     1322     1323     1324     1325     1326     1327     1328 
## 3.224873 3.678174 3.267317 3.267529 3.426740 3.426740 3.426740 3.426740 
##     1329     1330     1331     1332     1333     1334     1335     1336 
## 3.444014 3.340062 3.340062 3.281888 3.283726 3.260200 3.367467 3.439355 
##     1337     1338     1339     1340     1341     1342     1343     1344 
## 3.522855 3.522855 3.522855 3.368068 3.368068 3.368068 3.352235 3.368068 
##     1345     1346     1347     1348     1349     1350     1351     1352 
## 3.146167 3.284219 3.488345 3.384857 3.384857 3.422047 3.266638 3.390802 
##     1353     1354     1355     1356     1357     1358     1359     1360 
## 3.338263 3.338263 3.392893 3.492917 3.489845 3.375179 3.346120 3.144851 
##     1361     1362     1363     1364     1365     1366     1367     1368 
## 3.261890 3.273134 3.144851 3.276241 3.391328 3.324903 3.357136 3.399889 
##     1369     1370     1371     1372     1373     1374     1375     1376 
## 3.276486 3.438713 3.263899 3.300210 3.263899 3.293981 3.411073 3.369266 
##     1377     1378     1379     1380     1381     1382     1383     1384 
## 3.273186 3.642916 3.393923 3.360803 3.360803 3.401793 3.287460 3.287460 
##     1385     1386     1387     1388     1389     1390     1391     1392 
## 3.371054 3.277281 3.358360 3.358360 3.416235 3.439355 3.524816 3.285245 
##     1393     1394     1395     1396     1397     1398     1399     1400 
## 3.395954 3.290544 3.445415 3.272338 3.271773 3.353460 3.401793 3.386506 
##     1401     1402     1403     1404     1405     1406     1407     1408 
## 3.288652 3.288652 3.332933 3.387593 3.283726 3.337666 3.289899 3.520915 
##     1409     1410     1411     1412     1413     1414     1415     1416 
## 3.291876 3.520915 3.415468 3.434933 3.289899 3.223451 3.209849 3.477222 
##     1417     1418     1419     1420     1421     1422     1423     1424 
## 3.209849 3.375179 3.330603 3.308251 3.330603 3.362021 3.416992 3.452271 
##     1425     1426     1427     1428     1429     1430     1431     1432 
## 3.281888 3.281888 3.335288 3.330603 3.319415 3.315200 3.365660 3.330603 
##     1433     1434     1435     1436     1437     1438     1439     1440 
## 3.556349 3.494490 3.196008 3.196008 3.220594 3.371054 3.413109 3.372830 
##     1441     1442     1443     1444     1445     1446     1447     1448 
## 3.384303 3.326593 3.416235 3.405469 3.372830 3.326593 3.416235 3.422047 
##     1449     1450     1451     1452     1453     1454     1455     1456 
## 3.420644 3.384303 3.384303 3.307306 3.439355 3.321582 3.139387 3.426740 
##     1457     1458     1459     1460     1461     1462     1463     1464 
## 3.504433 3.321582 3.276486 3.318345 3.462291 3.455236 3.422047 3.416992 
##     1465     1466     1467     1468     1469     1470     1471     1472 
## 3.420644 3.420644 3.359582 3.391852 3.359582 3.331765 3.217726 3.421349 
##     1473     1474     1475     1476     1477     1478     1479     1480 
## 3.308251 3.477222 3.210561 3.656249 3.210561 3.656249 3.368667 3.276741 
##     1481     1482     1483     1484     1485     1486     1487     1488 
## 3.576619 3.276741 3.332933 3.284725 3.396954 3.392374 3.404567 3.489845 
##     1489     1490     1491     1492     1493     1494     1495     1496 
## 3.592650 3.466496 3.392893 3.592650 3.462291 3.313156 3.474679 3.423420 
##     1497     1498     1499     1500     1501     1502     1503     1504 
## 3.313156 3.406361 3.417367 3.406361 3.339462 3.282781 3.363844 3.451458 
##     1505     1506     1507     1508     1509     1510     1511     1512 
## 3.318345 3.421349 3.407242 3.318345 3.375179 3.315200 3.456567 3.390802 
##     1513     1514     1515     1516     1517     1518     1519     1520 
## 3.437442 3.445415 3.377503 3.377503 3.492917 3.446856 3.331765 3.428031 
##     1521     1522     1523     1524     1525     1526     1527     1528 
## 3.446856 3.413905 3.492917 3.417740 3.502716 3.435558 3.429937 3.403653 
##     1529     1530     1531     1532     1533     1534     1535     1536 
## 3.296984 3.386506 3.444014 3.462799 3.387593 3.296212 3.433065 3.405469 
##     1537     1538     1539     1540     1541     1542     1543     1544 
## 3.486870 3.619731 3.460309 3.382068 3.474679 3.378655 3.414300 3.155736 
##     1545     1546     1547     1548     1549     1550     1551     1552 
## 3.274280 3.422737 3.413109 3.453942 3.152170 3.364450 3.395954 3.396954 
##     1553     1554     1555     1556     1557     1558     1559     1560 
## 3.454801 3.367467 3.421699 3.400846 3.418479 3.421699 3.387593 3.298569 
##     1561     1562     1563     1564     1565     1566     1567     1568 
## 3.298569 3.298569 3.368667 3.368667 3.368667 3.427387 3.419931 3.368667 
##     1569     1570     1571     1572     1573     1574     1575     1576 
## 3.404567 3.468723 3.433688 3.461291 3.334108 3.477222 3.489845 3.321582 
##     1577     1578     1579     1580     1581     1582     1583     1584 
## 3.281888 3.449873 3.410658 3.468723 3.362021 3.468723 3.461789 3.458402 
##     1585     1586     1587     1588     1589     1590     1591     1592 
## 3.422737 3.368068 3.344902 3.518995 3.348562 3.418111 3.453942 3.581191 
##     1593     1594     1595     1596     1597     1598     1599 
## 3.459346 3.419931 3.465414 3.513366 3.459346 3.493700 3.477222
Wine$pH.error.SVM <- (Wine$pH.Predict.SVM - Wine$pH)/Wine$pH

ggplot(data = Wine, aes(x = quality, y = pH.error.SVM)) +
  geom_boxplot(outlier.colour = 'red')

rmse(SVM$residuals)
## [1] 0.106785

Linear Model and Support Vector Machine

We see that a SVM functions slightly better than a LM.
- The first RMS is of the LM, the second of the SVM.
- The median % error hovered at or near zero for most wine qualities.
- Notably, wines rated with a quality of 3 had large negative error.
- We can interpret this finding by saying that for many of the ‘bad’ wines, total acidity from tartaric, acetic, and citric acids were a worse predictor of pH. Simply put, it is likely that there were other components–possibly impurities–that changed and affected the pH.

As annotated previously, we hypothesized that free.sulfur.dioxide and total.sulfur.dioxide were dependent on each other.
Plotting this:


ggplot(data = Wine, aes(x = free.sulfur.dioxide, y = total.sulfur.dioxide)) +
  geom_point(alpha=0.2) +
  geom_smooth(se=F)
## `geom_smooth()` using method = 'gam'

cor.test(Wine$free.sulfur.dioxide, Wine$total.sulfur.dioxide)
## 
##  Pearson's product-moment correlation
## 
## data:  Wine$free.sulfur.dioxide and Wine$total.sulfur.dioxide
## t = 35.84, df = 1597, p-value < 2.2e-16
## alternative hypothesis: true correlation is not equal to 0
## 95 percent confidence interval:
##  0.6395786 0.6939740
## sample estimates:
##       cor 
## 0.6676665


MULTIVARIATE PLOTS SECTION


ggplot(data = Wine,
       aes(x = citric.acid, y = volatile.acidity,
           color = rating)) +
  geom_jitter(alpha=1)

ggplot(data = Wine,
       aes(x = alcohol, y = log10(sulphates),
           color = rating)) +
  geom_point(alpha=0.2) 

ggplot(data = Wine,
       aes(x = pH, y = alcohol, color = rating)) +
  geom_point(alpha=0.2) 

ggplot(data = Wine,
       aes(x = pH, y = alcohol, color = rating)) +
  geom_point()


From the plots we come to know the following:


FINAL PLOTS AND SUMMARY


PLOT ONE: EFFECTS OF ACID ON WINE QUALITY


### Plot One: Effects of Acid on Wine Quality
ggplot(data = Wine, aes(x = quality, y = fixed.acidity,
fill = quality)) + 
ylab('Fixed Acidity (g/dm^3)') +
xlab('Quality') +
geom_violin()+guides(fill=F)

grid.arrange(ggplot(data = Wine, aes(x = quality, y = fixed.acidity,
fill = quality)) + 
ylab('Fixed Acidity (g/dm^3)') +
xlab('Quality') +
geom_violin()+guides(fill=F),
ggplot(data = Wine, aes(x = quality, y = volatile.acidity,
fill = quality)) +
ylab('Volatile Acidity (g/dm^3)') +
xlab('Quality') +
geom_violin()+guides(fill=F), 
ggplot(data = Wine, aes(x = quality, y = citric.acid,
fill = quality)) +
ylab('Citric Acid (g/dm^3)') +
xlab('Quality') +
geom_violin()+guides(fill=F), 
ggplot(data = Wine, aes(x = quality, y = pH,
fill = quality)) +
ylab('pH') +
xlab('Quality') +
geom_violin()+guides(fill=F),top='Effects of Acid on Wine Quality')


SUMMARY ONE


PLOT TWO: EFFECTS OF ALCOHOL ON WINE QUALITY


### Plot Two: Effect of Alcohol on Wine Quality
ggplot(data = Wine, aes(x = quality, y = alcohol,
fill = rating)) +
geom_boxplot(outlier.color = 'red') +
ggtitle('Alcohol Levels in Different Wine Qualities') +
xlab('Quality') +
ylab('Alcohol (% volume)')


SUMMARY TWO


PLOT THREE: What makes good wines, good, and bad wines, bad?


ggplot(data = subset(Wine, rating != 'average'),
aes(x = volatile.acidity, y = alcohol,
color = rating)) +
geom_point(alpha=0.4) +
ggtitle('Alcohol vs. Volatile Acidity and Wine Quality') +
xlab('Volatile Acidity (g / dm^3)') +
ylab('Alcohol (% volume)') + geom_smooth(method = 'lm',se=F)

plot_ly(Wine,x=~rating,y=~volatile.acidity,z=~alcohol,type='scatter3d')

SUMMARY THREE


MAIN CONCLUSION